A positive integer n has exactly two distinct prime factors, p and q, such that p + q = 12… — ACT

A positive integer $n$ has exactly two distinct prime factors, $p$ and $q$ , such that $p + q = 12$ . If $n$ has exactly 8 positive factors, what is the least possible value of $n$ ?

A
35
B
245
875Answer
D
1,225
E
1,715

Answer

875

The prime factors of the number must be 5 and 7 since they are the only prime numbers that sum to 12. For a number with prime factorization

5^a \times 7^b

, the number of factors is

(a+1)(b+1) = 8

. Since both prime factors must be present, the exponents must be 1 and 3. The two possible numbers are

5^3 \times 7^1 = 875

and

5^1 \times 7^3 = 1,715

. The least value is 875.

Step-by-Step Solution

Find the two prime numbers

p

and

q

that sum to 12.

The prime factors are 5 and 7.

Since

p

and

q

are prime and

p + q = 12

, we test pairs of positive integers. The only pair of prime numbers that sums to 12 is 5 and 7.

Express the number of factors of

n

using the prime factorization formula.

The number of factors is

(a + 1)(b + 1) = 8

For any positive integer

n = p^a \times q^b

, the number of positive factors is given by the formula

(a + 1)(b + 1)

Determine the possible values for the exponents

a

and

b

The set of exponents

\{a, b\}

must be

\{1, 3\}

Since

n

must contain both prime factors,

a \ge 1

and

b \ge 1

, which means

(a + 1) \ge 2

and

(b + 1) \ge 2

. The only factor pair of 8 where both factors are at least 2 is

2 \times 4

. Thus, one exponent must be 1 and the other must be 3.

Calculate the possible values of

n

and identify the least value.

The least value is 875.

The two possible values for

n

are

5^3 \times 7^1 = 125 \times 7 = 875

and

5^1 \times 7^3 = 5 \times 343 = 1,715

. The least possible value is 875.

Key Concept

Factors, Multiples, and Prime Factorization

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