Two positive integers m and n satisfy \text{GCD}(m, n) = 60 and \text{LCM}(m, n) = 75{}600… — GMAT

Two positive integers $m$ and $n$ satisfy $\text{GCD}(m, n) = 60$ and $\text{LCM}(m, n) = 75{}600$ . Given that $m$ is divisible by $63$ but not by $189$ , and $n$ is divisible by $400$ , what is the value of $m$ ?

Answer: 1260

Answer

1260

By finding the prime factorizations of the GCD (

2^2 \cdot 3^1 \cdot 5^1

) and LCM (

2^4 \cdot 3^3 \cdot 5^2 \cdot 7^1

), we determine the prime powers for

m

and

n

using

\min

and

\max

rules. The constraint that

n

is divisible by

400 = 2^4 \cdot 5^2

fixes

e_2(n)=4

and

e_5(n)=2

, which forces

e_2(m)=2

and

e_5(m)=1

. The constraint that

m

is divisible by

63 = 3^2 \cdot 7^1

but not

189 = 3^3 \cdot 7^1

fixes

e_3(m)=2

and

e_7(m)=1

. Evaluating

m = 2^2 \cdot 3^2 \cdot 5^1 \cdot 7^1

gives

1{}260

Step-by-Step Solution

Prime factorize the GCD and LCM

GCD = 2^2 * 3^1 * 5^1 * 7^0 and LCM = 2^4 * 3^3 * 5^2 * 7^1

GCD represents the minimum exponent of each prime shared by m and n, whereas LCM represents the maximum exponent.

Determine the prime exponents for m using divisibility conditions

e_2(m) = 2, e_3(m) = 2, e_5(m) = 1, e_7(m) = 1

Divisibility of n by 400 forces e_2(n)=4 and e_5(n)=2, leaving e_2(m)=2 and e_5(m)=1. Divisibility of m by 63 but not 189 fixes e_3(m)=2 and e_7(m)=1.

Compute the product of prime powers for m

m = 4 * 9 * 5 * 7 = 1260

Multiplying the determined prime factors yields the exact value of integer m.

Key Concept

Greatest Common Divisor (GCD) and Least Common Multiple (LCM) Prime Exponent Rules

Estimated Time:2m 0s

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