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Senior Manager Joined: 20 May 2014
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If k is an integer, what is the smallest possible value of k [#permalink] 00:00

Question Stats: 75% (01:00) correct 25% (01:00) wrong based on 20 sessions
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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[Reveal] Spoiler: OA VP Joined: 20 Apr 2016
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WE: Engineering (Energy and Utilities)
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Re: If k is an integer, what is the smallest possible value of k [#permalink]
1
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Bunuel wrote:
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

Kudos for correct solution.

Here let 1060 can be written as = $$2^4 X 65^1$$

to make it perfect square k should be equal = 65 i.e $$2^4 X 65^2$$

Hence option E
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Manager Joined: 29 Nov 2017
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GRE 1: Q142 V146 WE: Information Technology (Computer Software)
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Re: If k is an integer, what is the smallest possible value of k [#permalink]
I multipled each option to 1040 and took a square root by the calculator and I got the answer in 2.03 min .. E
Manager Joined: 26 Jan 2018
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GRE 1: Q165 V156 Followers: 1

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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
Manager Joined: 29 Nov 2017
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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.
Director Joined: 09 Nov 2018
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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.

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

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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Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html MyGuru Representative Joined: 09 Apr 2020
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Re: If k is a positive integer, what is the smallest possible va [#permalink]
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Expert's post
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

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Steve Markofsky Intern Joined: 24 Jan 2020
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Re: If k is a positive integer, what is the smallest possible va [#permalink]
3
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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 Re: If k is a positive integer, what is the smallest possible va   [#permalink] 18 Apr 2020, 05:00
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