For positive integers a, b, and c, let x = 2^a \times 3^3 \times 5^b and y = 2^3 \times 3^… — GMAT

For positive integers $a$ , $b$ , and $c$ , let $x = 2^a \times 3^3 \times 5^b$ and $y = 2^3 \times 3^c \times 5^1$ . If $\text{GCD}(x, y) = 360$ and $\text{LCM}(x, y) = 270,000$ , what is the value of $a + b + c$ ?

A
8
B
9
10Answer
D
11
E
12

Answer

First, express the given GCD and LCM in prime factor form:

360 = 2^3 \times 3^2 \times 5^1

and

270,000 = 2^4 \times 3^3 \times 5^4

. For any two numbers, the exponent of each prime factor in their GCD is the minimum of their individual exponents, while the exponent in their LCM is the maximum. Comparing prime 2:

\max(a, 3) = 4

, so

a = 4

. Comparing prime 3:

\min(3, c) = 2

, so

c = 2

. Comparing prime 5:

\max(b, 1) = 4

, so

b = 4

. Therefore,

a + b + c = 4 + 4 + 2 = 10

Step-by-Step Solution

Find the prime factorizations of GCD(x, y) and LCM(x, y)

360 = 2^3 × 3^2 × 5^1 and 270,000 = 2^4 × 3^3 × 5^4

Expressing GCD and LCM in prime factor form allows direct comparison of prime exponents.

Apply prime exponent rules for GCD (minimum) and LCM (maximum)

For prime 2: min(a, 3) = 3 and max(a, 3) = 4, so a = 4.
For prime 3: min(3, c) = 2 and max(3, c) = 3, so c = 2.
For prime 5: min(b, 1) = 1 and max(b, 1) = 4, so b = 4.

The GCD takes the minimum exponent for each prime factor, while the LCM takes the maximum exponent.

Calculate the sum a + b + c

4 + 4 + 2 = 10

Substitute the evaluated exponent values to find the requested total.

Key Concept

GCD and LCM via Prime Factorization Exponents

Estimated Time:1m 30s

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