Let A = 12^6 \cdot 35^4 and B = 18^4 \cdot 14^6. If d = \text{gcd}(A, B), how many positiv… — GMAT

Let $A = 12^6 \cdot 35^4$ and $B = 18^4 \cdot 14^6$ . If $d = \text{gcd}(A, B)$ , how many positive factors of $d^2$ are not factors of $d$ ?

Answer: 2072

Answer

The number of positive factors of

d^2

that are not factors of

d

is 2072.

Decomposing

A

and

B

into prime factors yields

A = 2^{12} \cdot 3^6 \cdot 5^4 \cdot 7^4

and

B = 2^{10} \cdot 3^8 \cdot 7^6

. Taking the minimum power of each common prime gives

d = \text{gcd}(A, B) = 2^{10} \cdot 3^6 \cdot 7^4

, which has

(10+1)(6+1)(4+1) = 385

positive factors. For

d^2 = 2^{20} \cdot 3^{12} \cdot 7^8

, the total number of positive factors is

(20+1)(12+1)(8+1) = 2457

. Subtracting the factors of

d

yields

2457 - 385 = 2072

Step-by-Step Solution

Find the prime factorizations of

A

and

B

A = 2^{12} \cdot 3^6 \cdot 5^4 \cdot 7^4

and

B = 2^{10} \cdot 3^8 \cdot 7^6

Converting composite bases into prime factors allows determination of common divisor properties.

Determine the greatest common divisor

d = \text{gcd}(A, B)

d = 2^{10} \cdot 3^6 \cdot 7^4

The GCD takes the minimum exponent for each common prime factor between

A

and

B

Calculate the total number of positive factors of

d

385 positive factors

Adding 1 to each prime exponent of

d

and multiplying gives

(10+1)(6+1)(4+1) = 385

Find the prime factorization and number of positive factors of

d^2

d^2 = 2^{20} \cdot 3^{12} \cdot 7^8

, which has 2457 positive factors

Squaring

d

doubles all prime exponents. The number of factors is

(20+1)(12+1)(8+1) = 2457

Subtract the number of factors of

d

from the number of factors of

d^2

2457 - 385 = 2072

Since every factor of

d

is also a factor of

d^2

, the factors of

d^2

that are not factors of

d

equal the total factors of

d^2

minus the factors of

d

Key Concept

Prime Factorization, Greatest Common Divisor (GCD), and the Divisor Count Formula

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