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# How many positive divisors does integer m have?

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How many positive divisors does integer m have? [#permalink]
BrentGMATPrepNow wrote:
How many positive divisors does positive integer m have?

(1) m³ has exactly 13 positive divisors
(2) m² has exactly 9 positive divisors

Solution:

We are asked the number of factors of positive integer m

if $$m=(p_1)^a\times (p_2)^b\times (p_3)^c.....$$, where $$p_1, p_2, p_3$$ and so on are distinct prime numbers, then total number of factors $$=(a+1)\times (b+1)\times (c+1).....$$

Statement 1: $$m^3$$ has exactly 13 positive divisors

Since 13 is a prime number, it can only be written as $$13\times 1$$ which when we compare with $$(a+1)\times (b+1)$$, we get $$a=12$$ and $$b=0$$

Thus, we can be sure that $$m^3=p^{12}$$ where p is a prime number
$$⇒m=p^{\frac{12}{3}}$$
$$⇒m=p^4$$

So, total factors of $$m=4+1=5$$

Thus, statement 1 alone is sufficient and we can eliminate options B, C and E

Statement 2: $$m^2$$ has exactly 9 positive divisors

9 can be written as either $$9\times 1$$ or $$3\times 3$$

in first case, $$a+1=9$$ or $$a=8$$ and $$m^2=p^8$$
In the second case, $$a+1=3$$ or $$a=2$$ and $$b+1=3$$ or $$b=2$$ and finally $$m^2=(p_1)^2\times (p_2)^2$$

In both these cases, we will get different total factors of m

Thus, statement 2 alone is not sufficient

Hence the right answer is Option A
How many positive divisors does integer m have? [#permalink]
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