Brent Hanneson wrote:

heshamelaziry wrote:Positive integer P has exactly 2 positive prime factors, 5 and 11. If P has a total of 8 positive factors, including 1 and P, what is the value of P?
A) 125 is a factor of P.
B) 121 is not a factor of P.

Great question.
First, there's a little rule you should know:
If the prime factorization of a number, n, is such that n = p^a * q^b * r^c etc where p, q, r (etc) are prime numbers, then the number of positive divisors (factors) that n has is (a+1)(b+1)(c+1) etc
(e.g., 600=2^3 * 3^1 * 5*2, so 600 has 24 positive divisors since(3+1)(1+1)(2+1)=24)

We are told that P has two unique prime factors (5 and 11), which means that we can write P as follows: P = 5^x * 11^y, where x and y are both positive integers.
Our goal here is to find the value of P. In other words, we need to find the values of x and y.

Since we are told that P has exactly 8 positive divisors, we know (from the above rule) that (x+1)(y+1)=8
Since x and y must be positive integers, there are only two possible sets of values that satisfy the equation (x+1)(y+1)=8. The two possible solutions are i) x=1 and y=3, and ii) x=3 and y=1.

(A) if 125 (aka 5^3) is a factor of P, then we can conclude that x is 3 or greater. This means that x=3 and y=1, since we can now rule out the solution x=1 and y=3. SUFFICIENT

The answer is D

(B) If 121 (aka 11^2) is not a divisor of P, we know that y must be less than 2. In other words y must equal 1, in which case our only possible solution is x=3 and y=1. SUFFICIENT

Brent Hanneson wrote:

heshamelaziry wrote:Positive integer P has exactly 2 positive prime factors, 5 and 11. If P has a total of 8 positive factors, including 1 and P, what is the value of P?
A) 125 is a factor of P.
B) 121 is not a factor of P.

Great question.
First, there's a little rule you should know:
If the prime factorization of a number, n, is such that n = p^a * q^b * r^c etc where p, q, r (etc) are prime numbers, then the number of positive divisors (factors) that n has is (a+1)(b+1)(c+1) etc
(e.g., 600=2^3 * 3^1 * 5*2, so 600 has 24 positive divisors since(3+1)(1+1)(2+1)=24)

We are told that P has two unique prime factors (5 and 11), which means that we can write P as follows: P = 5^x * 11^y, where x and y are both positive integers.
Our goal here is to find the value of P. In other words, we need to find the values of x and y.

Since we are told that P has exactly 8 positive divisors, we know (from the above rule) that (x+1)(y+1)=8
Since x and y must be positive integers, there are only two possible sets of values that satisfy the equation (x+1)(y+1)=8. The two possible solutions are i) x=1 and y=3, and ii) x=3 and y=1.

(A) if 125 (aka 5^3) is a factor of P, then we can conclude that x is 3 or greater. This means that x=3 and y=1, since we can now rule out the solution x=1 and y=3. SUFFICIENT

The answer is D

(B) If 121 (aka 11^2) is not a divisor of P, we know that y must be less than 2. In other words y must equal 1, in which case our only possible solution is x=3 and y=1. SUFFICIENT

heshamelaziry wrote:I do not understand the above solution. Seems too complicated one for number properties. Why should the factor be 5^x and 11^y ?