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142.) If y is the smallest positive integer such that 3150 multiplied by y is the square of an integer, then y must be

a) 2 b) 5 c) 6 d) 7 e) 14

upon factoring 3150 i got following prime factors 3,3,5,5 nd 2

3,150=2*3^2*5^2*7, now 3,150*y to be a perfect square y must complete the odd powers of 2 and 7 to even number (perfect square has even powers of its primes), so the least value of y is 2*7=14. In this case 3,150y=(2*3^2*5^2*7)*(2*7)=(2*3*5*7)^2=perfect square.

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23 Mar 2012, 21:27

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E.

Create a prime factor tree for 3,150 and see what numbers do not have a pair. 3150 = 5x5x3x3x7x2.... the 7 and 2 do not have a second pair to make it a perfect square of a number. so y must be 7*2 = 14
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15 Aug 2014, 14:24

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11 May 2016, 00:44

3150 = 3 * 1050 = 3*3*350 = 3*3*35*10 = 3*3*5*7*5*2 = (3)^2 * (5)^2 * 14 So we can say if we want this number to be a perfect square this has to be multiplied with 14 . y=14

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11 May 2016, 07:24

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GMATD11 wrote:

If y is the smallest positive integer such that 3,150 multiplied by y is the square of an integer, then y must be

(A) 2 (B) 5 (C) 6 (D) 7 (E) 14

Solution:

This problem is testing us on the rule that when we express a perfect square by its unique prime factors, every prime factor's exponent is an even number.

Let’s start by prime factorizing 3,150.

3,150 = 315 x 10 = 5 x 63 x 10 = 5 x 7 x 3 x 3 x 5 x 2

3,150 = 2^1 x 3^2 x 5^2 x 7^1

(Notice that the exponents of both 2 and 7 are not even numbers. This tells us that 3,150 itself is not a perfect square.)

We also are given that 3,150 multiplied by y is the square of an integer. We can write this as:

2^1 x 3^2 x 5^2 x 7^1 x y = square of an integer

According to our rule, we need all unique prime factors' exponents to be even numbers. Thus, we need one more 2 and one more 7. Therefore, y = 7 x 2 = 14

Answer is E.
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Jeffrey Miller Scott Woodbury-Stewart Founder and CEO

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