For two positive integers m and n, the greatest common divisor is \gcd(m, n) = 15 and the … — GRE General Test

For two positive integers $m$ and $n$ , the greatest common divisor is $\gcd(m, n) = 15$ and the least common multiple is $\text{lcm}(m, n) = 900$ . If $m = 75$ , what is the total number of positive divisors of $n$ ?

18Answer
B
9
C
5
D
12
E
3

Answer

The correct answer is 18. First, use the relation

m \cdot n = \gcd(m, n) \cdot \text{lcm}(m, n)

to find

n = \frac{15 \cdot 900}{75} = 180

. Next, write 180 in prime factorized form:

180 = 2^2 \cdot 3^2 \cdot 5^1

. The total number of positive divisors is found by adding 1 to each exponent and multiplying the results:

(2 + 1)(2 + 1)(1 + 1) = 3 \cdot 3 \cdot 2 = 18

Step-by-Step Solution

Calculate the value of

n

using the fundamental product identity for GCD and LCM.

n = \frac{\gcd(m, n) \cdot \text{lcm}(m, n)}{m} = \frac{15 \cdot 900}{75} = 180

For any two positive integers, the product of the integers equals the product of their GCD and LCM.

Find the prime factorization of

180

180 = 2^2 \cdot 3^2 \cdot 5^1

Breaking down 180 into prime powers allows determination of the total count of positive divisors.

Apply the divisor counting formula by adding 1 to each prime exponent and multiplying.

(2+1)(2+1)(1+1) = 3 \cdot 3 \cdot 2 = 18

If an integer has prime factorization

p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}

, the number of positive divisors is

(e_1+1)(e_2+1)\cdots(e_k+1)

Key Concept

GCD and LCM fundamental identity (

a \cdot b = \gcd(a,b) \cdot \text{lcm}(a,b)

) combined with the prime factorization divisor counting formula.

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

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