Two positive integers a and b have prime factorizations of the form a = 2^x \times 3^y \ti… — GRE General Test

Two positive integers $a$ and $b$ have prime factorizations of the form $a = 2^x \times 3^y \times 5^z$ and $b = 2^u \times 3^v \times 5^w$ , where $x, y, z, u, v, w$ are non-negative integers. The greatest common divisor (GCD) of $a$ and $b$ is $180$ , and the least common multiple (LCM) of $a$ and $b$ is $54,000$ . If $a$ has exactly $24$ positive divisors, what is the value of $b$ ?

Answer: 18000

Answer

18000

Prime factorizing

\text{GCD}(a,b) = 180 = 2^2 \times 3^2 \times 5^1

and

\text{LCM}(a,b) = 54,000 = 2^4 \times 3^3 \times 5^3

reveals that the pair of exponents for bases

2, 3, 5

are

\{2, 4\}

\{2, 3\}

, and

\{1, 3\}

respectively. Applying the total divisor formula

(x+1)(y+1)(z+1) = 24

for

a

x+1

must divide

24

. Since

x \in \{2, 4\}

x

must be

2

(since

5

does not divide

24

). This leaves

(y+1)(z+1) = 8

. With

y \in \{2, 3\}

y+1

must be

4

(since

3

does not divide

8

), giving

y = 3

and

z = 1

. Thus,

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

. The remaining exponent choices for

b

are

u = 4

v = 2

, and

w = 3

, giving

b = 2^4 \times 3^2 \times 5^3 = 18,000

Step-by-Step Solution

Factorize GCD and LCM into prime bases.

GCD = 2^2 * 3^2 * 5^1; LCM = 2^4 * 3^3 * 5^3.

GCD represents the minimum prime exponents and LCM represents the maximum prime exponents of a and b.

Set up candidate exponent pairs for primes 2, 3, and 5.

{x, u} = {2, 4}, {y, v} = {2, 3}, and {z, w} = {1, 3}.

For any prime p, min(exp_p(a), exp_p(b)) = exp_p(GCD) and max(exp_p(a), exp_p(b)) = exp_p(LCM).

Solve for x, y, and z using the divisor count formula d(a) = (x+1)(y+1)(z+1) = 24.

x = 2, y = 3, z = 1.

If x = 4, then x+1 = 5, which is not a factor of 24. Thus x = 2. Then (y+1)(z+1) = 8. Since y in {2, 3}, y+1 = 4 implies y = 3 and z = 1.

Determine the exponents of b and calculate b.

u = 4, v = 2, w = 3, so b = 2^4 * 3^2 * 5^3 = 18,000.

Complementary values from the sets {x,u}, {y,v}, and {z,w} give the exponents for b.

Key Concept

Relationship between prime factor exponents in GCD, LCM, and divisor counting functions.

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