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The function f(n) = the number of factors of n. If p and q a

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The function f(n) = the number of factors of n. If p and q a

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The function f(n) = the number of factors of n. If p and q are positive integers and f(pq) = 4, what is the value of p?

(1) p + q is an odd integer
(2) q is less than p

What's the best way to determine which statement is sufficient? Is there any experts can help?

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ardz24 wrote:
The function f(n) = the number of factors of n. If p and q are positive integers and f(pq) = 4, what is the value of p?

(1) p + q is an odd integer
(2) q is less than p
Target question: What is the value of p?

Given: f(n) = the number of factors of n. If p and q are positive integers and f(pq) = 4
IMPORTANT: Notice that the product pq will be sure to have AT LEAST 4 factors.
If p and q are PRIME numbers, then the product pq has the following factors: 1, p, q and pq
However, if p and q are NOT prime numbers, then there will be more than 4 factors, since p and q will each have more than 2 factors each
For example, if p = 4 and q = 6, then the product, 24, has more than 4 factors: 1, 2, 3, 4, 6, 8, 12, and 24

BIG TAKEAWAY: If the product pq has exactly 4 factors, then p and q are both prime numbers

Statement 1: p + q is an odd integer
We already know that p and q are both prime numbers
If 2 prime numbers add to be an ODD number, then we know that one of the numbers must be 2 and the other number must be an odd prime.

Here are two sets of values for p and q that satisfy statement 1:
Case a: p = 2 and q = 3. Notice that the product, 6, has 4 factors: 1, 2, 3 and 6. In this case, p = 2
Case b: p = 7 and q = 2. Notice that the product, 14, has 4 factors: 1, 2, 7 and 14. In this case, p = 7
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: q is less than p
From the given information, we know that p and q are both prime numbers.
Here are two sets of values for p and q that satisfy statement 2:
Case a: p = 3 and q = 2. Notice that the product, 6, has 4 factors: 1, 2, 3 and 6. In this case, p = 3
Case b: p = 7 and q = 2. Notice that the product, 14, has 4 factors: 1, 2, 7 and 14. In this case, p = 7
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
There are several values of p and q that satisfy BOTH statements. Here are two:
Case a: p = 3 and q = 2. Notice that the product, 6, has 4 factors: 1, 2, 3 and 6. In this case, p = 3
Case b: p = 7 and q = 2. Notice that the product, 14, has 4 factors: 1, 2, 7 and 14. In this case, p = 7
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,
Brent

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