Let m = 2^a \cdot 5^3 \cdot 11^b and n = 2^4 \cdot 5^c \cdot 11^2, where a, b, and c are p… — GMAT

Let $m = 2^a \cdot 5^3 \cdot 11^b$ and $n = 2^4 \cdot 5^c \cdot 11^2$ , where $a$ , $b$ , and $c$ are positive integers. If the greatest common divisor of $m$ and $n$ is $2^3 \cdot 5^2 \cdot 11^2$ , and the least common multiple of $m$ and $n$ has exactly $180$ positive integer divisors, what is the value of $a + b + c$ ?

Answer: 13

Answer

The value of

a + b + c

13

To find

a, b, c

, analyze prime factor exponents. The GCD uses minimum exponents:

\min(a,4)=3 \implies a=3

\min(3,c)=2 \implies c=2

, and

\min(b,2)=2 \implies b \ge 2

. The LCM uses maximum exponents:

2^4, 5^3, 11^b

. The number of divisors of the LCM is

(4+1)(3+1)(b+1) = 20(b+1) = 180

, yielding

b=8

. Adding

a+b+c = 3+8+2 = 13

Step-by-Step Solution

Use the prime exponent rule for greatest common divisor to find

a

and

c

, and bound

b

a = 3

c = 2

, and

b \ge 2

The greatest common divisor takes the minimum exponent for each prime factor:

\min(a, 4) = 3 \implies a = 3

\min(3, c) = 2 \implies c = 2

, and

\min(b, 2) = 2 \implies b \ge 2

Determine the prime factorization of

\text{lcm}(m, n)

\text{lcm}(m, n) = 2^4 \cdot 5^3 \cdot 11^b

The least common multiple takes the maximum exponent for each prime factor:

\max(3, 4) = 4

\max(3, 2) = 3

, and

\max(b, 2) = b

since

b \ge 2

Apply the total number of divisors formula to solve for

b

b = 8

The number of positive integer divisors of

2^4 \cdot 5^3 \cdot 11^b

(4+1)(3+1)(b+1) = 20(b+1) = 180

, which yields

b+1 = 9

b = 8

Calculate the sum

a + b + c

a + b + c = 13

Summing the values

a=3

b=8

, and

c=2

gives

3 + 8 + 2 = 13

Key Concept

GCD and LCM prime factor exponent rules and total divisor formula

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