The prime factorizations of two positive integers A and B are given by A = 2^a \times 3^5 … — GRE General Test

The prime factorizations of two positive integers $A$ and $B$ are given by $A = 2^a \times 3^5 \times 5^b$ and $B = 2^4 \times 3^c \times 7^2$ , where $a$ , $b$ , and $c$ are positive integers. If the greatest common divisor of $A$ and $B$ is $\gcd(A, B) = 2^2 \times 3^3$ , and their least common multiple is \text{lcm} $(A, B) = 2^4 \times 3^5 \times 5^3 \times 7^2$ , what is the value of $a + b + c$ ?

Answer: 8

Answer

The correct answer is 8.

For any two positive integers expressed in prime factorized form, the greatest common divisor contains each prime factor raised to the minimum of its exponents in the two numbers, while the least common multiple contains each prime factor raised to the maximum of its exponents. For prime factor 2, the GCD has exponent 2, so min(a, 4) = 2, giving a = 2. For prime factor 3, the GCD has exponent 3, so min(5, c) = 3, giving c = 3. For prime factor 5, the LCM has exponent 3, so max(b, 0) = 3, giving b = 3. Adding these values together yields a + b + c = 2 + 3 + 3 = 8.

Step-by-Step Solution

Analyze the prime factor

2

min(a, 4) = 2, so a = 2

The greatest common divisor takes the minimum exponent for each prime factor shared between A and B.

Analyze the prime factor

3

min(5, c) = 3, so c = 3

The exponent of 3 in the GCD is 3, which must equal the smaller of the two exponents 5 and c.

Analyze the prime factor

5

max(b, 0) = 3, so b = 3

The least common multiple takes the maximum exponent for each prime factor present in either number.

Calculate the sum

a + b + c

2 + 3 + 3 = 8

Add the solved values of the three unknown prime exponents.

Key Concept

Relating prime factor exponents to GCD (minimum exponent) and LCM (maximum exponent)

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic