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Find the smallest integer k such that 198k is a perfect square. I was unable to proceed after this.

Find the smallest integer k such that 198k is a perfect square. It should also be noted that . Find the smallest positive integer such that is an integer. Dec 29, 2010 · Here's a question I came across and am having problems solving. Jan 30, 2021 · The question is that, Find all primes $p$ such that $p!+p$ is a perfect square. I hope this will help you If you feel this answer is helpful mark me as brainliest please Express 198 as the product of its prime factors. Therefore, we can write: 96k = 2^6 * 3 * k For 96k to be a square number, k must contain an even power of 3. Problem Find the sum of all positive integers such that the base- integer is a perfect square and the base- integer is a perfect cube. Solution 1 The first step is to convert and into base-10 numbers. So, the smallest k for which 360k is a perfect square is k = 2 * 5 = 10. Cubed number calculator with steps. Therefore, this reduces to a telescope series: Thus, we need to be an integer; this can be only , which occur when is an integer. Submitted by Michael S. 2 B. Therefore, the greatest perfect square less than or equal to 244 is 15 2 = 225. In other words, the perfect squares are the squares of the whole numbers such as 1 or 1 2, 4 or 2 2, 9 or 3 2, 16 or 4 2, 25 or 5 2 and so on. Feb 23, 2025 · The smallest four-digit perfect square is 1024, which is the square of 32. It remains to prove that there exists at most one such integer. The task is to find the perfect square number closest to N and steps required to reach this number from N. We have to find the smallest possible value of k. We know that cannot be as isn't divisible by , so 1004 doesn't divide . It is To make $$\frac {126} {k}$$k126 a perfect square, we need to find the smallest value of $$k$$k such that when divided into 126, it will have two of each prime factor. Write the prime factors as pairs such that each pair has two same prime factors. Let k be a positive integer and 1350k is a square number. So, it is already in a perfect square form. 11^2 * 3^2 * 2^2 = 4356 The square root of Factor 200 as 2^3 * 5^2. The factors of 1350k can be written To find the greatest perfect square less than or equal to 244, take the square root of 244 and round off to its nearest integer. n is the product of the factors that appear an odd number of times in the array list. 900 is a perfect square. \cdot18!$ is a perfect square. I also tried another approach We can see that for p=2,3 the given expression is a perfect square For each even positive integer , let denote the greatest power of 2 that divides For example, and For each positive integer let Find the greatest integer less than 1000 such that is a perfect square. Feb 8, 2023 · The smallest positive integer k such that 1350k is a square number is equals to the 6. The highest common factor of x and 198 is 33. The following steps will be useful to find the least number which has to multiplied by the given number to get a perfect square. If \ (N\) is an integer, then \ (N^2\) is a perfect square. Jul 20, 2023 · Let $n$ be the smallest positive integer such that n is divisible by $20$, $n^2$ is a perfect cube, and $n^3$ is a perfect square. 441 The smallest 5-digit integer perfect square is 10,000 = (100)2The largest 5-digit integer perfect square is 99,856 = (316)2So we want to know how many numbers that is, from 100 to 316 inclusive. Step 2 : Definitely the given value must be greater than the May 21, 2022 · A square number, perfect square, or simply "a square" is what is produced when you multiply an integer (a "whole" number), positive, negative, or zero, times itself. $63=9\cdot 7$ is ruled out, since $7$ would be the smaller possible choice. Jan 19, 2025 · To find the smallest positive integer k such that 540k is a perfect cube, we need to ensure that all the prime factors of 540k have exponents that are multiples of 3. Q11. Example: 3 x 3 = 9 Thus: 9 is a perfect square. Is the proof correct? To find the smallest positive integer $$n$$n such that $$150n$$150n is a perfect square, we need to find the square root of $$150n$$150n and see if it is an integer. c) Find the smallest possible integer value of k such that 198k is a square. Expressed as the product of prime factors. A perfect square is an integer that can be expressed as the product of two equal integers. Question: (b) Find the smallest possible positive integer k such that 504×k is a perfect square. So, the smallest possible value of integer n is 5 × 7 = 35. 360/3^1 = 120 Now, we need to find the smallest positive integer k such that 120/k is a square number. For an integer to be a perfect square, each prime factor has to occur an even number of time. Feb. 5 C. Then, we can write and . Problem If are consecutive positive integers such that is a perfect square and is a perfect cube, what is the smallest possible value of ? Solution Since the middle term of an arithmetic progression with an odd number of terms is the average of the series, we know and . Aug 18, 2023 · Expressed as the product of prime factors, 198 = 2 x 32 x 11 and 90 = 2 x 32 x 5. Dec 6, 2023 · The question asks to find the smallest positive integer 'n' such that the product of 140 and 'n' is the square of an integer. Use these results to find (a) the smallest integer, k, such that 198k is a perfect square, Sep 7, 2023 · To find the smallest integer k such that multiplication of 198 and k is a perfect square and 54 and k is a perfect cube, the values are 11 and 2 respectively. To make $$360k$$360k a perfect cube, we need to find the smallest $$k$$k such that $$k$$k contains the same prime factors as $$360$$360, but with their exponents increased by a multiple of 3. Found 4 solutions by mszlmb, Prithwis, lyra, amit5562: Answer by mszlmb (115) (Show Source): To do this, we need to find the smallest possible values of the exponents of 2 and 3 that will make the product a square. Please help me in this question. On multiplying both sides by 4, we get $ The best we can do is $ (ab)^2\times b$, but $b$ is not in the squared term, therefore it is not a perfect square (remember, we assumed $b$ is not a perfect square). Sep 6, 2023 · Find the smallest positive integer n such that 2n is a perfect square, 3n is a perfect cube, and 5n is a perfect fifth power. Doing the division, we find that 96 = 3 * 32. Now from the quadratic formula, Because is an integer, this means for some nonnegative integer . Find the smallest integer $a$ such that $P = a\cdot1!\cdot2!\cdot3!\cdot4!\cdot5!. Find po Jan 31, 2021 · For the smallest positive integer k such that 180 k is a perfect square, we first factorize 180 to its prime factors: 180 = 22 × 32 × 51. Q1) a) Find the smallest possible value of k such that 270k is a cube number. We are left with $7$ and Need someone to help check my proof, the reason why i assumed n = k is because the original question states that if n is a perfect square, then n + 2 is not a perfect square. Jul 8, 2022 · The smallest positive integer n such that 3n is a perfect square and 2n is a perfect cube is 108. It is given that, 120k is a perfect square. Feb 20, 2016 · Can someone help me with this exercise? I tried to do it, but it was very hard to solve it. We want to find the value of $n$ in $2^5\times 3\times 5^2\times 7^3\times n$ so that the latter expression will be a perfect square. 2. To find the smallest integer value of P such that PN K, where K is a perfect square, we start by analyzing the prime factorization of N: N = 24 × 31 × 75 For K to be a perfect square, all prime factors must have even exponents. Therefore k would need to be the product of whatever primes occur in 96 an odd number of times. Examples: Input: N = 100 Output: 4 Explanation: There are four factors of 100 (1, 4, 25, 100) that are perfect square. Completing the square, we have Apr 28, 2022 · To find the smallest positive integer ( n ) such that ( 2n ) is a perfect square, ( 3n ) is a perfect cube, and ( 4n ) is a perfect fourth, we analyze the conditions for each case using prime factorization. Hint: Store all smallest factors of m into an array list. Sample Problems Find the smallest positive integer k such that 504k is a perfect square. Oct 19, 2019 · Answer: 22 Step-by-step explanation: factor 198 into prime numbers 198 = 2 * 99 = 2 * 3 * 33 = 2 * 3 * 3 * 11 = 2 * 3^2 * 11 For a perfect square, each prime factor will be even. Can you give the answer? Find the smallest positive integer k such that 96k is a square number. Jan 23, 2017 · Comparing this to $kx=wy$ tells us that there is some integer number $\ell$ such that $k=5\ell$ and $w=9\ell$ Now take $x+y=2z$ to get $5x+5y=10z\Rightarrow 14y=10z$, or $7y=5z$ after dividing by $2$. Jun 28, 2016 · What is the smallest positive integer K such that 126*k is the square of a positive integer? A. The smallest multiple of that that is a perfect cube is $2^6\cdot 3^6\cdot 7^3\cdot 11^3$ which means it must be multiplied by $2^2\cdot 3^1\cdot 7^2 Friends, I want the answer for this. What's reputation and how do I get it? Instead, you can save this post to reference later. $$ So the solution is all those pairs of integers $ (a,b)$ such that $b = 5m$ for some integer $m$. Jul 28, 2025 · A number is a perfect square if all the exponents in its prime factorization are multiples of 2. Some key properties include: Perfect Squares are the only numbers that have odd number of distinct divisors. n is the product of the factors that appear an odd number of times in the array list. (c) x is a number between 200 and 300. Therefore, we can write Find the smallest positive integer k k, such that product of 420 420 and k k is a perfect square. Solution The prime factorization of 504 is 504 = 23 32 7. Aug 5, 2023 · In order for 140n to be a perfect square, n must contain at least one additional 5 and at least one additional 7. Input: N = 900 Output: 8 Explanation: There are eight factors of 900 (1, 4, 9, 25, 36, 100, 225, 900) that are perfect square. Jul 23, 2025 · A number is a perfect square if it can be expressed as n2 , where "n" is an integer. We have to find the smallest positive number such that 1050 becomes a perfect square. To check the least number to be subtracted to make the number as perfect square, we have to follow the steps. May 8, 2023 · To find the smallest positive integer k such that 315 times k is a perfect square, we need to look at the prime factorization of 315. Thus or , giving or . Mar 16, 2023 · Your All-in-One Learning Portal: GeeksforGeeks is a comprehensive educational platform that empowers learners across domains-spanning computer science and programming, school education, upskilling, commerce, software tools, competitive exams, and more. k = 11 * 3 <<<<< ======= answer. 8 Find the smallest positive integer value of k such that 945k is a perfect cube Singapore Math Tutors 1K subscribers Subscribe Now this only gives you an integer if $n = 2m$ is even, and in that case you get $$ b = 5m. (It is therefore correct when you note that $a$ can be any integer). Find the smallest positive integer to be multiplied to get a #perfect square Maths help for students 883 subscribers Subscribed The least number with four digits is 1000. 💡 Integer is a name for a whole number that can be either negative, positive, or equal to 0. Since $1$ is not among the choices given, let's assume that $1575$ is not a square from the beginning. Perfect Squares from 1 to 100 Below shows the list of perfect squares from 1 to 100 along with their factors (product Aug 24, 2019 · Answer: k = 15 Step-by-step explanation: 60 times some number k must be a perfect square. For example, 2016 is squarish, because the nearest perfect square to 2016 is 452 = 2025 and 2025 2016 = 9 is a perfect square. (ii) Find the smallest positive integer k such that 594/sqrt (k) is a perfect cube. This is found by determining the prime factorization of 450 and adjusting the exponents to be multiples of 3. 264k = 132 2k = 11 12 k = 11 3 2 2*k To get a perfect square, you are going to have to have an even number of prime factors. The smallest possible value of this sum is Solution Solution 1 Let be the consecutive positive integers. Jun 3, 2018 · Similarly, $4^ {973}= (2^ {973})^2$ and the next biggest square is $ (2^ {973}+1)^2 = 4 {973}+2^ {974}+1$, so we must have $$4 {973}+2^ {974}+1 \leq 4^ {973}+4^ {x-27}+1$$ Feb 29, 2024 · The smallest possible integer k such that the product 1575 * k is a perfect square can be found by prime factorizing 1575 and ensuring each prime factor has an even power in the final product. If n is represented in its factorized form as pa ⋅ qb ⋅ rc ⋅ …, where p,q,r,… are all distinct prime numbers, then find the minimum sum a+ b + c + …. You have one 11 and one 3. Their sum, , is a perfect square. Find step-by-step Computer science solutions and the answer to the textbook question (Algebra: perfect square) Write a program that prompts the user to enter an integer $\mathrm {m}$ and find the smallest integer $\mathrm {n}$ such that $\mathrm {m} { }^* \mathrm {n}$ is a perfect square. Problem Find the sum of all positive integers for which is a perfect square. Solution 2 Suppose there is some such that . This conclusion is based on ensuring that n contains the necessary powers of 2 and 3. Aug 24, 2016 · Integers Lillianah E. Dec 5, 2024 · Find an answer to your question Find the smallest integer value of 'k' where 168k is a perfect square. Co Perfect Squares and their Square Roots Perfect Square: Taking a positive integer and squaring it (multiplying it by itself) equals a perfect square. Jun 30, 2023 · The smallest positive integer that Joelle could have multiplied 756 by to obtain a perfect square is 3. b) Find the least common multiple (LCM) of 180, 90, and 140. This is calculated by finding the square root of 1000, rounding it up to the nearest whole number, and squaring that number. To determine this, we need to factorize 140 and see which smallest positive integer when multiplied by 140 will make it a perfect square. Taking the square root (principal square root) of that perfect square equals the original positive integer. Answer x= _ [2] Click here 👆 to get an answer to your question ️ Write down the smallest positive integer, k, such that 150k is a perfect square. Therefore, if 504k is a perfect square, then k must be of the form 2 7 m where m is a perfect square. For that, we have to find the square root of 1000 by the long division method as shown below: 1000 is 24 (124 − 100) less than the nearest square number 32 2. To find the smallest possible integer k such that 96k is a square number, we need to factorize 96 into its prime factors. (Hint: Store all smallest factors of m into an array list. 144 D. Solution Solution 1 By the product-to-sum identities, we have that . For the prime factor 3: The Feb 18, 2023 · The smallest positive integer value of K that makes 450K a perfect square is 2, since 450 needs another factor of 2 in its prime factorization to become a perfect square. Find the smallest positive integer n such that n/2 is a perfect square, n/3 is a perfect cube and n/5 is a perfect fifth power. [University discrete math] Find the least positive integer k such that m*k is a perfect cube? Let m = 2 4 * 3 5 * 7 * 11 2 What positive value does k have to be for the product of mk to be a perfect cube? I know that if I'm looking for a perfect square, I take the non-squared bases and multiply them together, but what about perfect cubes? Mar 3, 2018 · Try factoring this first. (c) the smallest positive integer value of ‘n’ for which 168n is a multiple of 324. b) For 360k to be a perfect cube, the exponents of all prime factors in its prime factorization must be multiples of 3. For 200k to be a square number, k must be a multiple of 3 and 2, and it must be a perfect square. multiply it by 2 and 11 so that you get 2^2 * 3^2 * 11^2 so k = 2 * 11 = 22 Jun 27, 2025 · Examples Perfect squares are useful in various real-life situations, such as calculating areas and volumes. Step 1 : Using long division, find the out the nearest perfect square of the given number. Find the smallest possible value of x. Perfect Square Numbers We know that the square of a number is that number times itself. The missing exponents for 2 and 5 are 2 and 1 respectively, so multiplying by k=20 achieves this. Sep 23, 2017 · The smallest positive integer k such that 1350k is a perfect cube is found by prime factorizing 1350 and ensuring that all prime exponents in its factorization are multiples of 3. To find the smallest positive integer that Joelle could have multiplied 756 by to obtain a perfect square, let's break down the prime factorization of 756. By using Wilson's Theorem I got, $$p!+p=p ( (p-1)!+1)=p^ {2}k$$ If $p^ {2}k$ is a perfect square, then k must be a perfect square. Therefore, multiple checks against the conditions confirm that n = 120 is the correct answer. Input: N = 3 Output: 100 961 Approach: For increasing values of N starting from N = 1, the series will go on like 9, 81, 961, 9801, . Dec 8, 2024 · k = 98. It's 316 minus the first 99 = 217 of them. Megan uses all 594 cubes to make a cuboid. 196 E. This is determined by ensuring all prime factor exponents in the factorization of 96k are even. For the exponent of 2, we need an even number since a square of an odd number will not have a factor of 2. For example, when we multiply 3 × 3 = 3², results 9. So all square numbers are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on. This is found by ensuring all prime factors in the resulting quotient's prime factorization are raised to multiples of 3. Mar 16, 2022 · The least positive integer k such that 96k is a perfect square is k = 6. The number 525k is a perfect square. We need one more 2, one more 3, and one more 7 to make the exponents even. Express 594 as a product of its prime factors. Enter any non negative integer number in the input field of the calculator and get the answer. We can factor 120 as 2^3 * 3 * 5. Not the question you’re looking for? Post any question and get expert help quickly. 756 = 2^2 x 3^3 x 7 Thus the smallest perfect cube that is a multiple of 756 is 2^3 × 3^3 × 7^3; to obtain this need to multiply 756 by 2^1 × 3^0 × 7^2 = 98 Thus the smallest k to make 756k a perfect prime is k = 98. What will be the smallest number k such that if we concatenate the digits of n with those of k we get a perfect square? For example, for n=1 the smallest k is 6 since May 31, 2023 · The smallest positive integer n such that 2n is a perfect square, 3n is a perfect cube, and 5n is a 5th power is n = 120. Sep 17, 2019 · In mathematics, to find the smallest positive integer, k, such that 3240/k is a square number, we need to prime factorize 3240 first. The other is to find the prime factors of 96. Hint • Store all smallest factors of m into an array list. Step 1: Factorize 264. Dec 1, 2022 · Consider a positive integer n. Any help would be greatly appreciated! What is the smallest positive integer K such that the product of 1575 x K is a perfect square? A perfect square number is a number that can be created by multiplying two identical integers; in other words, the perfect square's root is a whole number. 10 D. Therefore, it is clear that is the smallest Aug 11, 2019 · Find the smallest positive integer n&amp;gt;10 such that n+6 is a prime and 9n+7 is a perfect square Get the answers you need, now! Aug 19, 2024 · To find the smallest integer value K such that 200K is both a square number and a multiple of 60, we start by analyzing the prime factorizations of the numbers involved. Calculate the maximum number Hence, there exists at least one positive integer $k$ such that $S / k!$ is a perfect square, namely $k = n/2$. Learn about perfect square numbers in this article along with examples of perfect squares, important tips, and examples. The given number is 1050. (b) The number 198k is a perfect square. Computer Science questions and answers (Algebra: perfect square) Write a program that prompts the user to enter an integer m and find the smallest integer n such that m * n is a perfect square. Jun 20, 2018 · Show that there are in nitely many Diophantine quadruples: posi-tive integers (a1; a2; a3; a4) such that aiaj + 1 is a perfect square for all 1 i < j 4. Given that 9+6 is divisible by 3, we know that 96 is divisible by 3. (Of the positive integers between 1 and 10, o s between 1 and N, inclusive. Find the prime factor which does not occur in pair. m. 65­ Ep. for the largest N-digit perfect Question: (Algebra: perfect square) Write a program that prompts the user to enter an integer m and find the smallest integer n such that m*n is a perfect square. The prime factorization of a perfect square must include each prime factor an even number of times. To determine the smallest positive integer value of K so that 450K is a perfect square, we must factor 450 to identify which prime factors are not squared and then multiply by the least amount of such factors to make them The question does not provide enough information to determine the smallest positive integer k. 22, 2022 11:26 p. Mar 31, 2015 · If k is an integer, what is the smallest possible value of k such that 1040*k is the square of an integer? A. Jan 30, 2021 · The question is as follows Determine all integers n such that $n^ {4}-n^ {2}+64$ is the square of an integer Here is my approch Let $n^ {4}-n^ {2}+64=k^ {2}$. Find the smallest positive integer value o k. Examples: Input: N = 2 Output: 16 81 16 and 18 are the smallest and the largest 2-digit perfect squares. The smallest possible such perfect square is when , and the sum is . To find the least square number with four digits, we must find the smallest number that must be added to 1000 in order to make a perfect square. Problem 9. For instance, the square root of 25 is 5. 96 = 2^5 * 3 Since a square number has an even power of each prime factor, we need to add one more factor of 2 to make the power of 2 even. 6 (Kiran Kedlaya, PEN?). Because of this definition, perfect squares are always non-negative. Question ens. Aug 19, 2024 · Given that 200 = 23 × 52, Find the smallest integer value k such that 200k is both a square number and a multiple of 60. 15 E. In the prime factorization (in power format) of a perfect cube, every prime must be to the power of a multiple of 3. The prime factors of 3240 are 2³, 3⁴, and 5. By determining the necessary conditions for each case using prime factorization, we conclude that 120 satisfies these mathematical requirements. So at first, write the factors of a number 1050. 244 ≈ 15. Since is a perfect square, it follows that is a perfect square. Find all positive integers n such that 32n + 3n2 + 7 is a perfect square. Properties of Perfect Squares Perfect squares have unique characteristics that differentiate them from other numbers. Smallest Positive Integer - PRMO 2019 Find the smallest positive integer n ≥ 10 such that n+6 is a prime and 9n+7 is a perfect square. A perfect square is a number that can be expressed as the product of an integer with itself. Computer Science questions and answers Algebra: perfect square) Write a program that prompts the user to enter an integer m and find the smallest integer n such that m * n is a perfect square. Upvoting indicates when questions and answers are useful. Use these results to find the smallest integer, k, such that 198k is a perfect square. To find the smallest integer, k, such that 198k is a perfect square, we need to start with the prime factorization of 198. The highest common factor of x and 126 is 21. Home Mathematics Given That 198 = 2 X 3 X 11 And 90 = 2 X 3 X 5,a) Find The Smallest Integer, K, Such That 198k Is A Perfect MathematicsHigh School Given that 198 = 2 x 3² x 11 and 90 = 2 x 3² x 5, a) find the smallest integer, k, such that 198k is a perfect square b) the largest integer that is a factor of both 198 and 90 (pls show working (b) the largest integer which is factors of both 168 and 324. Feb 11, 2025 · The smallest integer k,such that 108k is a perfect square number Get the answers you need, now! Jul 4, 2023 · 𝒏 is the smallest positive integer such that 𝒏/𝟐 is a perfect square, 𝒏/𝟑 is a perfect cube and 𝒏/𝟓 is a perfect fifth power. Problem The sum of consecutive positive integers is a perfect square. Find the value of $n$ to make $2^8 + 2^{11} + 2^n$ a perfect square. Click here 👆 to get an answer to your question ️ a) The smallest integer, k, such that 198k is a perfect square, Dec 9, 2022 · Learn how to find smallest value of integer k such that product becomes perfect square or cubeConcept related to square and cube roots Oct 11, 2010 · Find the smallest positive integer value of n for which 168n is a multiple of 324. (c) Each cube has a volume of 2cm^3. Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, Question 32179: Find the smallest positive integer n such that 2n is a perfect square, 3n is a perfect cube, and 5n is a perfect fifth power. If we want to divide 3240 by some positive integer k to get a square number, we need the resultant number to still have all its prime factors raised to an even So, we need to divide 3^1 from 360. Currently, the exponent of 2 is 3, which is already a multiple of 3. 1. Given an integer n, you need to find the minimum number of perfect square numbers that add up to exactly n. In general, for a product to be a perfect square, all the prime factors in its prime factorization need to occur an even number of times. When a number is multiplied by the same number, the resultant number is called as a square number. Apr 7, 2024 · The smallest positive integer value of k such that 450k is a perfect cube is 60. The prime factorization of 315 is 32 x 5 x 7. 264 = 23 ×31 × 111 Step 2: For 264k to be a perfect square, all primes in its factorization must have even exponents. So, 9 is a square number. May 26, 2024 · Answer: The smallest integer is 5 180 must be multiplied by 5 to get perfect square. Here's how to solve it step by step: Perfect squares are the squares of the integers, or the product of an integer with itself. This is a multiple-choice problem for a time-tight exam so I need to be as fast as possible. A perfect square has even powers of prime factors, so we only need to multiply by an extra 5 to achieve this, making the smallest value of k equal to 5. Each of the sides of the cuboid is made up of more than 3 cubes. Factor 60 to see which values do not have a pair: 60 = 2 · 2 · 3 · 5 --> 60 = 2 · 2 · 3 · 5 ↓ 3 · 5 are not pairs So, you need to multiply 60 by 15 to make a perfect square 60 × 15 = 900 → 900 = 30 Jan 1, 2021 · The smallest positive integer k such that 96k is a square number is 6. I was unable to proceed after this. What is the number of digits of $n$ ? Jul 11, 2025 · Given a positive integer N N . 222350455212623504=233+32×7504+7=24 Answer (c) Write down the cube of 315 as a product of its prime factors in index notation. For example, if you are designing a square garden with an area of 198k square meters and you want the side length to be an integer, you need to find the smallest k that makes 198k a perfect square. 1050 = 2 × 3 × 7 × 5 × 5 The exponent of 5 is 1, which is odd, so we need to multiply 360 by another 5. 14 B. If you find the square root of a number and it’s a whole integer, that tells you that the number is a perfect square. Square numbers are always positive. We need to find a k that makes 1125 k a perfect cube and 196/ k a perfect square integer. The smallest positive value of m which is a (Algebra: perfect square) Write a program that prompts the user to enter an integer m and finds the smallest integer n such that m ⋅n is a perfect square. Online perfect calculator with solution steps and descriptions. Find the number of cubes on each side of the cuboid. Naive Approach: The simplest approach to solve this Jul 18, 2021 · A A perfect square is a number that can be termed as the product of two equal integers. So I'll assume n = k to proceed. Similarly, a perfect cube is an integer that can be expressed as the product of three equal Oct 9, 2023 · The smallest **positive **integer value of k such that 84k is a perfect square is 21, because you need to multiply 84 by a number which includes at least one more '3' and one more '7', the smallest of which is 21. Feb 1, 2015 · Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, If $1575\times k$ is a perfect square, we can write it as $\prod_k p_k^2$. Solution 2 Notice that all five choices given are (b) Find the smallest possible positive integer k such that 504×k is a perfect square. Learn the definition, formula, list, tips and tricks, facts with examples. Thus, 322 = 1024. This gives or , and the sum is . Click here 👆 to get an answer to your question ️ The number 504k is a perfect cube. Now put that into 264k and you get 11 * 3 * 2 * 2 * 11 * 3 which is a perfect square. Decompose the given numbers into its prime factors. The smallest integer k that satisfies the condition is k = 3 * 2^2 = 12. 36 C. Find the prime factorization of 168: 168 = 2³ * 3 * 7. Examples: Input: N = 1500 Output: Perfect square = 1521, Steps = 21 For N = 1500 Dec 9, 2022 · Learn how to find smallest value of integer k such that product becomes perfect square or cubeConcept related to square and cube roots dj p(j)(k) = p(x)jx=k dxj ) at k) i t square is a perfect square. A perfect square must have all prime factors raised to an even power. This online perfect square calculator tells you whether or not any given number is a perfect square. Jun 15, 2022 · Given an integer N, the task is to find the smallest and the largest N digit numbers which are also perfect squares. Let’s examine each prime factor: For the prime factor 2: The exponent is 4, which is even. Therefore, k = 2 * 3 * 7 = 42 Finding Smallest Integer To Multiply to a number to make it a perfect square Wai Zin 83 subscribers Subscribed Dec 20, 2020 · Explanation To find the smallest integer value of k for which 264k is a perfect square, we first need to look at the prime factorization of 264. Both conditions of being a perfect square and a perfect cube imply specific requirements for the prime factorization of n. Here you are given the number $2^4\cdot 3^5\cdot 7^1\cdot 11^2$. 6205 Rounding off to the nearest integer, get 15. 3. Answer k= _ [1] (c) x is a number between 200 and 300. (b) Megan is playing with 594 cubes. (Algebra: perfect square ) Write a program that prompts the user to enter an integer m and find the smallest integer n such that m * n is a perfect square. Note: The closest perfect square to N can be either less than, equal to or greater than N and steps are referred to as the difference between N and the closest perfect square. Find step-by-step Computer science solutions and the answer to the textbook question (Algebra: perfect square) Write a program that prompts the user to enter an integer m and find the smallest integer **n** such that **m*n** is a perfect square. To make 168k a perfect square, each prime factor must appear an even number of times. asked • 08/24/16 Find the smallest positive integer find the smallest positive integer,N, such that the product of 135 and N is a perfect square. You are going to have to do something about those two. Solution 1 If for some positive integer , then rearranging we get . ------------------ Expressed as the product of prime factors, 198 = 2 x 3^2 x 11 and 90 = 2 x 3^2 x 5. Square Root Calculator . The correct option is b) 3. Mar 8, 2021 · In order to be a "perfect cube" the prime factorization of a number must be primes all to a power a multiple of 3. Find the smallest positive integer value of k. This is determined by prime factorizing 96 and adding the necessary factors to make it a perfect square. Question: Write a program that prompts the user to enter an integer m and find the smallest integer n such that m * n is a perfect square. Therefore $9=3^2$ and $25=5^2$ are ruled out, since multiplying a non square with a square doesn't make it a square. That is the least number to be multiplied by the given number to get a Aug 22, 2024 · The smallest positive integer k such that k126 is a cube number is k = 126. For example, $2^3= 8$ is a perfect cube but so is $2^6= 64= 4^3$. Because there are less perfect cubes than perfect squares for the restriction we are given on , it is best to list out all the perfect Problem Let . For example, \ (100\) is a perfect square because it is equal to \ (10\times 10\). Jul 15, 2025 · Given an integer N, the task is to find the number of factors of N which are a perfect square. Friends, I want the answer for this. Rearranging gives . A perfect square is a number that can be expressed as the product of an integer by itself or as the second exponent of an integer. Feb 27, 2024 · By examining the prime factorization of the integer n given the conditions that it must be divisible by 20, that 2n is a perfect cube, and that 3n is a perfect square, we determine the smallest n is 2^3 × 3^2 × 5^1 = 360, which contains three digits. . Also, get the perfect square calculator here. 198=2 × 32 × 11 and 90 = 2 × 32 × 5 use these results to find (a) The smallest integer, k, such that 198k is a perfect square, To find the smallest positive integer value of k such that 198k is a perfect square number, we need to factorize 198 into its prime factors and then determine the value of k. Thus . The number 126k is a perfect square. Perfect cubed number checker. 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