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What is the smallest positive integer K such that the

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Joined: 19 Dec 2010
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What is the smallest positive integer K such that the  [#permalink]

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29 Dec 2010, 14:36
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What is the smallest positive integer K such that the product of 1575 x K is a perfect square?

A. 7
B. 9
C. 15
D. 25
E. 63

Hello,

Here's a question I came across and am having problems solving. Any help would be greatly appreciated!

Thanks for the help.
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Joined: 02 Sep 2009
Posts: 64249

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29 Dec 2010, 14:53
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m990540 wrote:
Hello,

Here's a question I came across and am having problems solving. Any help would be greatly appreciated!

What is the smallest positive integer K such that the product of 1575 x K is a perfect square?

A 7
B 9
C 15
D 25
E 63

Thanks for the help.

A perfect square, is just an integer that can be written as the square of some other integer. For example 16=4^2, is a perfect square.

Now, 1575 = 3^2 * 5^2 * 7, so if k=7 then 1575k = (3 * 5 * 7)^2, which is a perfect square (basically the least positive value of k must complete only the power of 7 to even power as powers of other primes are already even).

Check Number Theory chapter of Math Book for more on this topic: math-number-theory-88376.html

Questions about a perfect square for practice:
divisibility-103168.html
perfect-square-101678.html
a-perfect-square-79108.html

Hope it helps.
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29 Dec 2010, 17:22
Awesome, thanks so much for the help. Those links are great too.
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01 Jan 2011, 23:15
In a perfect square, all the prime constituents have to be raised to an even number. For example, 16 is perfect square which is equal to 2 raised to the power 4. 81 is a perfect square which includes 3 raised to the power 4.

In this example, the prime factors of 1575 are 3, 5, 7. 1575 can be written as a product of all its factors as: $$3^2*5^2*7^1$$. So you see, only 7 is having an odd number of powers. if the number gets one more 7, it will can be written as $$(3*5*7)^2$$, which is a perfect square. Answer is A. 7

If all prime factors are raised to the same even power, then the number is definitely a perfect square. In this example, if 7 is multiplied to 1575, all prime factors become raised to the power 2.

As Bunuel did, I will recommend you to check out the GMAT math book of this forum. To supplement this, you may try out MGMAT number properties. It covers all such required fundamentals in a very effective way.
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Re: What is the smallest positive integer...  [#permalink]

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19 Feb 2012, 18:39
What is the smallest positive integer k such that the product 1575*k is a perfect square?

1575 * k = 3 * 3 * 5 * 5 * 7 * k

for perfect square , k should be at least 7. Hence A.
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Re: What is the smallest positive integer K such that the  [#permalink]

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06 Jan 2017, 07:59
m990540 wrote:
What is the smallest positive integer K such that the product of 1575 x K is a perfect square?

A. 7
B. 9
C. 15
D. 25
E. 63

Hello,

Here's a question I came across and am having problems solving. Any help would be greatly appreciated!

Thanks for the help.

$$1575$$ = $$3^2$$ x $$5^2$$ x $$7^1$$

So , we are just 7 short of a perfect square number ..

Hence, correct answer must be (A) 7
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Re: What is the smallest positive integer K such that the  [#permalink]

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27 Jan 2018, 20:51
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: What is the smallest positive integer K such that the   [#permalink] 27 Jan 2018, 20:51