imane81 wrote:Please, need your help on this one. Thanks in advance
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 (greater than 1) 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:
E
