# If A has a total of 8 positive factors, including 1 and A

If A has a total of 8 positive factors, including 1 and A, what is the value of A? Positive integer A has exactly 2 positive prime factors, 5 and 11.

1. 125 is a factor of P
2. 121 is not a factor of P

A.Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B.Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C.BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D.EACH statement ALONE is sufficient.
E.Statements (1) and (2) TOGETHER are NOT sufficient.

Answer: Statement 1 tells us that P is divisible by 125. Since the prompt is concerned with the
prime factors of P, it will help to rewrite 125 as 5^3. Since P is divisible by 5^3, it is also divisible by 5^2 and 5. At this point, we have all 8 of P's factors: 1, 5, 5^2, 5^3, 11, 11 × 5, 11 × 5^2, and 11 × 5^3. Since P is a factor of itself, we know that the largest of these 8 factors, 11 × 5^3, must be equal to P.

Statement 1 is sufficient.

Statement 2 tells us that P is not divisible by 121, which we can write as 11^2. Since P's
only prime factors are 5 and 11, and since 11^2 is not a factor of P, we know that the remaining factors must either have only 5 as a prime factor or else be the product of 11 and a number that has only 5 as a prime factor. If we now include 1, 5, and 11, we see that we have identified exactly 8 factors. Again, the largest factor, 5^3 × 11, must be equal to P.

Statement 2 is also sufficient.

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