Video explanation [PQID: PS13829]: How many integers between 1 and 16, inclusive, have exactly three different positive…

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Video explanation [PQID: PS13829]: How many integers between 1 and 16, inclusive, have exactly three different positive…

R S says

How did you interpret that the question is asking about (p*2,p,1), when the question states How many …..integers have exactly three +ve integers. Why are 16,9 and 8 not included in the answer?

GMAT Quantum says

Hi,

16 has five factors: 1, 2, 4, 8, and 16.

9 does have three factors: 1, 3, and 9.

8 has four factors: 1, 2, 4, and 8.

We know prime numbers have only two factors, 1 and the number itself. We need numbers with three factors. If we look at numbers of the form $(p_1)(p_2)$, meaning product of two prime numbers, then we have four factors: $1, p_1, p_2,$ and $(p_1)(p_2)$. The only case left with three factors is $p^2$, a square of a prime number, which has only three factors: $1, p,$ and $p^2$.

I hope this makes sense.