If P is a positive integer and p^3 is divisible by 144

amit.lohchab wrote:1. If P is a positive integer and p^3 is divisible by 144, then the largest positive integer that must divide p is
A. 2
B. 3
C. 4
D. 6
E. 12

Ans E

144 = (2)(2)(2)(2)(3)(3)
So, we already know that the prime factorization of p must have at least one 2 and one 3 in it. This already means the correct answer is either D or E.

If p = 6, then p = (2)(3)
So, p^3 = (2)(2)(2)(3)(3)(3)
Since 144 = (2)(2)(2)(2)(3)(3), we can see that p^3 is NOT divisible by 144
So, the correct answer is NOT D

At this point, we can select E

To prove this, let's see what happens when p = 12
If p = 12, then p = (2)(2)(3)
So, p^3 = (2)(2)(2)(2)(2)(2)(3)(3)(3)
Since 144 = (2)(2)(2)(2)(3)(3), we can see that p^3 IS divisible by 144
So, it's possible that p = 12, which means the largest positive integer that must divide p is [spoiler]12 (answer choice E)[/spoiler]

Cheers,
Brent

amit.lohchab wrote:1. If P is a positive integer and p^3 is divisible by 144, then the largest positive integer that must divide p is

Let us assume, p³ = 144k = (12²)*k
We can see that, minimum possible value of k such that (12²)*k becomes a cube of an integer is 12.
Hence, minimum possible value of p is 12.

The correct answer is E.

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