What is the smallest integer K such that the product of 1575 x K is a perfect square?
A. 7
B. 9
C. 15
D. 25
E. 63
The OA is A.
Here's a question I came across and am having problems solving. Any expert that can help me with it, would be greatly appreciated! Thanks.
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What is the smallest positive integer K such that the...
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Hi swerve,
We're asked to find the SMALLEST integer K such that (1575)(K) is a perfect square. This question can be solved with Prime Factorization.
A 'perfect square' means that we're multiplying an integer by itself. For example:
(2)(2) = 4
(10)(10) = 100
In the second example, we can 'break down' the 10s into prime numbers...
(2x5)(2x5) = 100
The same concept applies here: we have to use the smallest K possible to create two equal terms....
(1575)(K) =
(5)(315)(K) =
(5)(5)(63)(K) =
(5)(5)(7)(9)(K) =
(5)(5)(7)(3)(3)(K)
Based on the above prime factorization, each of the two terms will have to include one 3, one 5 and one 7....
(3x5x7)(3x5xK)
Thus, K the smallest possible value of K is 7.
Final Answer: A
GMAT assassins aren't born, they're made,
Rich
We're asked to find the SMALLEST integer K such that (1575)(K) is a perfect square. This question can be solved with Prime Factorization.
A 'perfect square' means that we're multiplying an integer by itself. For example:
(2)(2) = 4
(10)(10) = 100
In the second example, we can 'break down' the 10s into prime numbers...
(2x5)(2x5) = 100
The same concept applies here: we have to use the smallest K possible to create two equal terms....
(1575)(K) =
(5)(315)(K) =
(5)(5)(63)(K) =
(5)(5)(7)(9)(K) =
(5)(5)(7)(3)(3)(K)
Based on the above prime factorization, each of the two terms will have to include one 3, one 5 and one 7....
(3x5x7)(3x5xK)
Thus, K the smallest possible value of K is 7.
Final Answer: A
GMAT assassins aren't born, they're made,
Rich
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GMATWisdom
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here we have to find with how many perfect square numbers i.e. 4,9,16,25,... it is divisible.swerve wrote:What is the smallest integer K such that the product of 1575 x K is a perfect square?
A. 7
B. 9
C. 15
D. 25
E. 63
The OA is A.
Here's a question I came across and am having problems solving. Any expert that can help me with it, would be greatly appreciated! Thanks.
any number whose sum of digits is divible by 9 must itself be divisible by 9.
if we add the digits of the number we get 1+5+7+5=18 which is a multiple of 9 so it must be divisible by 9.
dividing with 9 we get
(1575 x k)/9 = 175 x k
any figure which ends in 25, or 50, or 75 or 00 must be divisible by 25
further dividing with 25 we get = 7 x k
therefore k must be equal to 7 to make it a perfect square.
hence 7 is the smallest number to make it a perfect square.
so correct answer is A
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Here's a trick: break 1575K into two identical square roots.
For instance, say we have 36. 36 = 2 * 2 * 3 * 3 = (2 * 3) * (2 * 3), or two identical square roots.
Now let's try that with 1575K.
1575K =>
25 * 63 * K =>
5 * 5 * 3 * 3 * 7 * K =>
(5 * 3 * 7) * (5 * 3 * K)
So if K = 7, we'll have two identical roots. Touchdown!
For instance, say we have 36. 36 = 2 * 2 * 3 * 3 = (2 * 3) * (2 * 3), or two identical square roots.
Now let's try that with 1575K.
1575K =>
25 * 63 * K =>
5 * 5 * 3 * 3 * 7 * K =>
(5 * 3 * 7) * (5 * 3 * K)
So if K = 7, we'll have two identical roots. Touchdown!
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Matt@VeritasPrep
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- Joined: Wed Sep 12, 2012 3:32 pm
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Another approach (not quite as good, but you never know what's helpful!): pull the integers roots out of 1575K.
√1575K =>
√1575 * √K =>
√(25*9*7) * √K =>
√25 * √9 * √7 *√K =>
5 * 3 * √7 * √K =>
5 * 3 * √7K
So if √7K = an integer, we're set. If K = 7, then √7K = √49 = 7, an integer, so K = 7 is a solution.
√1575K =>
√1575 * √K =>
√(25*9*7) * √K =>
√25 * √9 * √7 *√K =>
5 * 3 * √7 * √K =>
5 * 3 * √7K
So if √7K = an integer, we're set. If K = 7, then √7K = √49 = 7, an integer, so K = 7 is a solution.
