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# If k is an integer, what is the smallest possible value of k such that

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Math Expert
Joined: 02 Sep 2009
Posts: 52294
If k is an integer, what is the smallest possible value of k such that  [#permalink]

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31 Mar 2015, 04:52
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Question Stats:

78% (01:22) correct 22% (01:43) wrong based on 251 sessions

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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

Kudos for a correct solution.

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Re: If k is an integer, what is the smallest possible value of k such that  [#permalink]

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31 Mar 2015, 05:08
1
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 a correct solution.

1040=2^4*13*5

We need 13*5 to be multiplied with 1040 to make is a perfect square.

Therefore, k=13*5=65

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Joined: 06 Nov 2014
Posts: 1877
Re: If k is an integer, what is the smallest possible value of k such that  [#permalink]

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31 Mar 2015, 10:36
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 a correct solution.

1040 = (2^4)*(13)*(5)
We need one more factor of 13 as well as 5 so as to make square of an integer.
So, k = 13*5 = 65
Hence option (E).

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Joined: 07 Aug 2011
Posts: 533
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Re: If k is an integer, what is the smallest possible value of k such that  [#permalink]

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31 Mar 2015, 10:48
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 a correct solution.

1040K = 13*8*10*K = 13*2^4*5*K
so K should have one 13 and one 5 i.e. 65 Answer E
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If k is an integer, what is the smallest possible value of k such that  [#permalink]

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31 Mar 2015, 11:21
1
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 a correct solution.

We need to find a number k such that 1040k = n^2 for some number n.

The prime factorization of 1040 is $$2^4 5^1 13^1$$.

In order for 1040k to be a square number, we need every prime factor in 1040k to be at least raised to the second power. The only numbers that are not are 5 and 13 - thus, we need one more factor of 5 and one more factor of 13. Therefore, k = 5*13 = 65.

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Re: If k is an integer, what is the smallest possible value of k such that  [#permalink]

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31 Mar 2015, 17:45
1

We require to find factors of 1040 which are perfect squares

1040 = 4 * 4 * 65

To make 1040 a perfect square, it has to be multiplied by 65

k = 65
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Joined: 29 Mar 2013
Posts: 26
Schools: ISB '16
Re: If k is an integer, what is the smallest possible value of k such that  [#permalink]

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31 Mar 2015, 20:07
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 a correct solution.

1040k - k needs to be replaced.

__1_02________
1 |1040K
1
-------------
202 | *040k
404
K should be 4
102^2 = 10404

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Re: If k is an integer, what is the smallest possible value of k such that  [#permalink]

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01 Apr 2015, 00:40
1
Factorizing 1040 = (2^4).5.13
Thus we need 5 and 13 to make it a perfect sqaure.
Thus 65. Ans E.
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Posts: 52294
Re: If k is an integer, what is the smallest possible value of k such that  [#permalink]

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06 Apr 2015, 04:52
1
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 a correct solution.

MAGOOSH OFFICIAL SOLUTION:
Attachment:

Making_A_Square.png [ 17.87 KiB | Viewed 6811 times ]

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Re: If k is an integer, what is the smallest possible value of k such that  [#permalink]

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19 Aug 2016, 03:51
Factorize 1040
1040 = 2 * 2 * 2 * 2 * 5 * 13
as we know that square of any number should have even powers of factors, the only number missing is 5 * 13 = 65
Intern
Joined: 15 Feb 2018
Posts: 6
Re: If k is an integer, what is the smallest possible value of k such that  [#permalink]

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10 Jun 2018, 10:59
1040
^
10*104
^ ^
2*5 52*2
^
26*2
^
13*2
2*2*2*2*5*13=1040
2*2*2*2*5*13*K=1040k
(2*2*5*13)=(2*2*K) Since its a square, distribute the 2's and set the sides equal
5*13 = K (Cancel out the 2's)
65=K
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Re: If k is an integer, what is the smallest possible value of k such that  [#permalink]

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12 Jun 2018, 03:26
Answer E and here is why,
First we need to factorize 1040, or 2^4*5*13*k=a perfect square. To have a perfect square, we must have squared variables before equal sign or 5 and 13. So 5*13 is 65
Re: If k is an integer, what is the smallest possible value of k such that &nbs [#permalink] 12 Jun 2018, 03:26
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