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

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Re: If k is an even integer, what is the smallest possible value of k [#permalink]
ScottTargetTestPrep wrote:
stonecold wrote:
If k is an even integer, what is the smallest possible value of k such that 3675*k is the square of an integer?

A.3
B.9
C.12
D.15
E.20

If 3675*k is a perfect square, then its prime factorization must contain only even exponents. Let’s begin by prime factoring 3,675.

3675 = 25 x 147 = 5 x 5 x 49 x 3 = 5 x 5 x 7 x 7 x 3 = 5^2 x 7^2 x 3^1

We can see that 3,675 is not a perfect square because its prime factorization contains an odd exponent (that is, 3^1). If k only has to be an integer, then the smallest value k can be is 3, since 3675*k would be 5^2 x 7^2 x 3^2, a perfect square. However, since the requirement is that k must be an even integer, we need k to be divisible by an even perfect square (notice that 3675 doesn’t have any even prime factors). Since the smallest even perfect square is 2^2 = 4, the smallest possible value of k is 3 x 4 = 12, so that 3,675*k is a perfect square.
In fact, if k = 12, then 3,675*k = 5^2 x 7^2 x 3^2 x 2^2, which is a perfect square.

[Note: The smallest possible value of k such that 3675*k is the square of an integer is actually 0, since 3675*0 = 0, which is 0^2. To avoid this case, the problem should be stated as “If k is a positive even integer…such that 3675*k is the square of a positive integer?”]

Absolutely Spot On.!!
P.S=> Edited the Question.
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Re: If k is an even integer, what is the smallest possible value of k [#permalink]
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stonecold wrote:
If k is a positive even integer, what is the smallest possible value of k such that 3675*k is the square of an integer?

A.3
B.9
C.12
D.15
E.20

$$3675 = 3^1 * 5^2 * 7^2$$

Now , is the most important part of this question

K must be even ( negate options A, B and D ) W e are left with C and E

Try one to confirm the answer...

Let's try (E)

$$20 = 2^2 * 5^1$$

$$3675*20 = 2^2 * 3^1 * 5^3 * 7^2$$ ( Can not square of an Integer )

Double check to be sure -

$$3675*12 = 2^2 * 3^2 * 5^2 * 7^2$$ (Square of an Integer )

Hope this helps !!

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Re: If k is an even integer, what is the smallest possible value of k [#permalink]
Abhishek009 wrote:
stonecold wrote:
If k is a positive even integer, what is the smallest possible value of k such that 3675*k is the square of an integer?

A.3
B.9
C.12
D.15
E.20

$$3675 = 3^1 * 5^2 * 7^2$$

Now , is the most important part of this question

K must be even ( negate options A, B and D ) W e are left with C and E

Try one to confirm the answer...

Let's try (E)

$$20 = 2^2 * 5^1$$

$$3675*20 = 2^2 * 3^1 * 5^3 * 7^2$$ ( Can not square of an Integer )

Double check to be sure -

$$3675*12 = 2^2 * 3^2 * 5^2 * 7^2$$ (Square of an Integer )

Hope this helps !!

You didn't fall for it this time

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Re: If k is an even integer, what is the smallest possible value of k [#permalink]
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stonecold wrote:
If k is a positive even integer, what is the smallest possible value of k such that 3675*k is the square of an integer?

A.3
B.9
C.12
D.15
E.20

I feel like killing myself for choosing A.
If k is a positive even integer
EVEN digit...grrr....
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If k is an even integer, what is the smallest possible value of k [#permalink]
stonecold wrote:
If k is a positive even integer, what is the smallest possible value of k such that 3675*k is the square of an integer?

A.3
B.9
C.12
D.15
E.20

3675=35*105
if k were odd, then 3 would give us 3*35*105=105^2
so we need an even multiple of 3
12 is only option
4*3*35*105)=(2*105)^2
12
C
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Re: If k is an even integer, what is the smallest possible value of k [#permalink]
Here are my 2 cents on this one ->

Given info => K is an even integer.
3675*k=square of an integer.

In a perfect square the power of each prime factor is an even number.
Breaking down 3675*k=> 5^2*7^2*3*k => Perfect square.
Now the general expression for k will be 3*(Any-prime)^even
If the question had not stated that k is even => A would have been sufficient.
But here as k is even => Minimum value of k=> 3*2^2 (why 2^2? => Beacuse 2 is the smallest positive even number )
Hence k=>2^2*3=>12

Hence C.
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Re: If k is an even integer, what is the smallest possible value of k [#permalink]
Ugh, I fell into the trap of forgetting that k is even, and chose A: k=3. I need to remember to always double check the constraints!
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If k is an even integer, what is the smallest possible value of k [#permalink]
sealberg wrote:
Ugh, I fell into the trap of forgetting that k is even, and chose A: k=3. I need to remember to always double check the constraints!

We all did my friend. We all did.

P.S -> Here is the Link to the entire Mock series -> https://gmatclub.com/forum/stonecold-s- ... 60-40.html
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If k is an even integer, what is the smallest possible value of k [#permalink]
It is mentioned that K is even integer so eliminate options A, B, D,

so consider option c-12*3675=44100 which is a perfect square of 210.

Ans is- C-12
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Re: If k is an even integer, what is the smallest possible value of k [#permalink]
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Re: If k is an even integer, what is the smallest possible value of k [#permalink]
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