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

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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

by M7MBA » Thu Jul 22, 2021 7:38 am

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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 $$p.$$

Source: Manhattan GMAT

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### Re: The function $$f(n) =$$ the number of factors of $$n.$$ If $$p$$ and $$q$$ are positive integers and $$f(pq) = 4,$$

by [email protected] » Fri Jul 23, 2021 5:27 am

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## Global Stats

M7MBA wrote:
Thu Jul 22, 2021 7:38 am
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 $$p.$$

Source: Manhattan GMAT
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

Statement 1: p + q is an odd integer
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
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