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Re: If k is an integer, what is the smallest possible value of k [#permalink]
65 as that is what is left post factoring. All the factors should have 2 values atleast
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Re: If k is an integer, what is the smallest possible value of k [#permalink]
One should also remember that the exponents of prime factors are even.

or we can solve the problem by taking the prime factorization of 1040 , when we do so we notice the factors of 13 and 5 are not even rather they are only 1 each..

hence we choose the option he because when we multiply 1040 by 65 13 and 5 factors make the exponents of 1040*65 even


hence option is E is correct.
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If k is an integer, what is the smallest possible value of k [#permalink]
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If k is an integer, what is the smallest possible value of k such that 1040k is the square of an integer?

A. 2

B. 5

C. 10

D. 15

E. 65
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Re: If k is an integer, what is the smallest possible value of k [#permalink]
1
1040 can be broken down into its component prime number as 2^4*5*13
the square of \(2^4\) is \(2^2\) however 65 does not have a integer square root therefore k has to be 65 as only then can we get a integer when a square root is taken
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Re: If k is an integer, what is the smallest possible value of k [#permalink]
IshanGre wrote:
One should also remember that the exponents of prime factors are even.



Please explain.
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Re: If k is an integer, what is the smallest possible value of k [#permalink]
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AE wrote:
IshanGre wrote:
One should also remember that the exponents of prime factors are even.



Please explain.


Since we are looking for square, the exponent has to be even..
Similarly if we are looking at a cube, the exponent should be divisible by 3..
.1040k=2*520k=\(2^2*260k=2^2*2*130k=2^4*65=2^4*5*13\)
So 2 has a power of 4, and we require one more of 5 and 13 to make the entire term as square
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Re: If k is a positive integer, what is the smallest possible va [#permalink]
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workout wrote:
If k is a positive integer, what is the smallest possible value of k such that 1040k is the square of an integer?

A) 2

B) 5

C) 10

D) 15

E) 65


Questions like this can often be solved by figuring out prime factors.

We know that 1040 * k is a perfect square, so find the prime factorization of 1040 first:

104*10
2*52* 5*2
2*2*26*5*2
2*2*2*13*5*2

So the prime factorization of 1040 = \(2^4*5*13\)

For a number to be a perfect square, each of its prime factors need to be paired with a matching prime factor.

\(2^4\) is 16, a perfect square. Each factor of 2 is paired with another factor of 2. But 5 and 13 don't have matching factors, so we need another 5 * 13 to make a perfect square. That product is k.

k = 5*13 = 65

Answer: E
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Re: If k is a positive integer, what is the smallest possible va [#permalink]
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Prime factorization of 1040 = \(2^4*5*13\)

Since \(\sqrt{2^4}\) = 4

Since \(2^4\) is already a perfect square the other factors not having a perfect square are 5 and 13

Hence the lowest number required to be multiplied to 1040 to make it a square = 5 * 13 = 65

Hence answer is E
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Re: If k is an integer, what is the smallest possible value of k [#permalink]
Use backsolving: Plug the answer choices in as k using the on-screen calculator and then take the square root of the product.
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Re: If k is an integer, what is the smallest possible value of k [#permalink]
Hi

Does the on-screen calculator have square-root? If yes, doesn't this make it a non-question!
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Re: If k is an integer, what is the smallest possible value of k [#permalink]
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It has
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Re: If k is an integer, what is the smallest possible value of k [#permalink]
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lindseyloo312 wrote:
Use backsolving: Plug the answer choices in as k using the on-screen calculator and then take the square root of the product.


Keep in mind that the onscreen GRE calculator can handle numbers less than 100,000,000
So, this question could be extended (using bigger numbers) to render the calculator useless.
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Re: If k is an integer, what is the smallest possible value of k [#permalink]
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Re: If k is an integer, what is the smallest possible value of k [#permalink]
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