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

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Bunuel
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If k is an integer, what is the smallest possible value of k such that [#permalink] New post   31 Mar 2015, 05:52
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If k is an integer, what is the smallest possible value of k such that 1040*k is the square of an integer?

A. 2
B. 5
C. 10
D. 15
E. 65­
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E
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AmoyV
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Re: If k is an integer, what is the smallest possible value of k such that [#permalink] New post   31 Mar 2015, 06:08
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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 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

Answer : E
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Re: If k is an integer, what is the smallest possible value of k such that [#permalink] New post   31 Mar 2015, 11:36
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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 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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Re: If k is an integer, what is the smallest possible value of k such that [#permalink] New post   31 Mar 2015, 11:48
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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Re: If k is an integer, what is the smallest possible value of k such that [#permalink] New post   31 Mar 2015, 12:21
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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 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.

Answer: E
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Re: If k is an integer, what is the smallest possible value of k such that [#permalink] New post   31 Mar 2015, 18:45
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Answer = E. 65

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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Re: If k is an integer, what is the smallest possible value of k such that [#permalink] New post   31 Mar 2015, 21:07
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


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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] New post   01 Apr 2015, 01:40
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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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Re: If k is an integer, what is the smallest possible value of k such that [#permalink] New post   06 Apr 2015, 05:52
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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 a correct solution.


MAGOOSH OFFICIAL SOLUTION:
Attachment:
Making_A_Square.png
Making_A_Square.png [ 17.87 KiB | Viewed 24687 times ]
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Re: If k is an integer, what is the smallest possible value of k such that [#permalink] New post   19 Aug 2016, 04:51
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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
Correct answer - E
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Re: If k is an integer, what is the smallest possible value of k such that [#permalink] New post   10 Jun 2018, 11:59
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] New post   12 Jun 2018, 04:26
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
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Re: If k is an integer, what is the smallest possible value of k such that [#permalink] New post   04 May 2019, 10:28
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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 a correct solution.


Since you need the 'square', so start by taking the square root (which will be equal to the integer we want to find since int*int=1040k)

Now,
(1040k)^1/2

= (2^4 * 13 * 5k)^1/2

= 4 (13 * 5k)^1/2

To make this value an integer,
you need to multiply it with a 13 and a 5 inside the root

therefore, k= 13* 5 = 65
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Re: If k is an integer, what is the smallest possible value of k such that [#permalink] New post   06 Sep 2019, 22:06
rhine29388 wrote:
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
Correct answer - E


"as we know that square of any number should have even powers of factors"

Can someone give me a link where this rule is discussed in more detail?

(if there even is such a thing)

For example what would I need to do if the number was "1080k"

Prime factorization would be: 3^3 * 2^3 * 5

So I have to add one 3, one 2 and one 5 that all factors are even? Is my thought approach right?

Thank you very much for any input
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Re: If k is an integer, what is the smallest possible value of k such that [#permalink] New post   15 Oct 2019, 13:17
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*2*2*2*5*13
So we need 5 and 13 to make it a perfect square. Which is 65
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Re: If k is an integer, what is the smallest possible value of k such that [#permalink] New post   16 Mar 2024, 10:00
Asked: If k is an integer, what is the smallest possible value of k such that 1040*k is the square of an integer?
 
1040 = 2^4*5*13
Smallest integer value of k = 5*13 = 65
1040*65 = 2^4*5^2*13^2 =( 2^2*5*13)^2

IMO E
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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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