Let N = 2^4 \times 3^2 \times 5. How many distinct prime factors does the integer N + 9 ha… — GMAT

Let $N = 2^4 \times 3^2 \times 5$ . How many distinct prime factors does the integer $N + 9$ have?

The integer

N + 9

has 1 distinct prime factor.

Evaluating

N = 2^4 \times 3^2 \times 5

gives

720

. Adding 9 yields

729

, which prime factorizes as

3^6

. Because 3 is the only unique prime base in this expression,

N + 9

has exactly 1 distinct prime factor.

Calculate the numerical value of

N

from its prime factorization.

N = 2^4 \times 3^2 \times 5 = 16 \times 9 \times 5 = 720

Evaluating

N

is necessary to compute

N + 9

Add 9 to

N

to find the value of

N + 9

N + 9 = 720 + 9 = 729

The question asks for the distinct prime factors of the sum

N + 9

Find the prime factorization of 729.

729 = 3^6

Prime factorization reveals the prime bases of the integer.

Count the number of distinct prime factors of 729.

The only distinct prime factor is 3, so the count is 1.

Distinct prime factors are the unique prime numbers appearing in the prime factorization.

Prime Factorization of an Expression Sum

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