For how many positive two-digit integers n is the sum n + (n + 2) equal to the product of … — GMAT

For how many positive two-digit integers $n$ is the sum $n + (n + 2)$ equal to the product of exactly two distinct prime numbers?

There are 21 positive two-digit integers that satisfy the given condition.

Simplifying

n + (n + 2)

gives

2(n + 1)

. For

2(n + 1)

to equal the product of two distinct prime numbers,

n + 1

must be an odd prime number

p

(so that 2 and

p

are distinct). Because

n

is a two-digit positive integer (

10 \le n \le 99

p = n + 1

must satisfy

11 \le p \le 100

. There are 25 prime numbers under 100, and exactly 4 of them (2, 3, 5, 7) are less than 11. Therefore, there are

25 - 4 = 21

suitable prime numbers, yielding 21 valid values of

n

Algebraically simplify the given sum.

n + (n + 2) = 2n + 2 = 2(n + 1)

Combining like terms isolates the common factor of 2.

Analyze the prime factorization structure.

The expression

2(n + 1)

has 2 as a prime factor. For it to be the product of exactly two distinct prime numbers,

n + 1

must be a prime number

p

distinct from 2 (i.e., an odd prime).

n + 1

were composite or equal to 2, the total number of distinct prime factors would not be exactly two distinct primes.

Determine the valid range for the prime

p = n + 1

Since

n

is a two-digit positive integer,

10 \le n \le 99

. Adding 1 to all parts gives

11 \le n + 1 \le 100

, so

11 \le p \le 100

The constraints on

n

dictate the bounds for the prime number

p

Count the number of prime numbers in the range

[11, 100]

There are 25 prime numbers less than 100. The primes less than 11 are 2, 3, 5, and 7 (4 primes). Thus, there are

25 - 4 = 21

primes in the range

[11, 100]

Subtracting primes outside the valid range yields the count of valid values for

n

Prime Factorization and Prime Number Properties

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