A positive integer n has exactly three distinct prime factors. If n^2 has exactly 147 posi… — GMAT

A positive integer $n$ has exactly three distinct prime factors. If $n^2$ has exactly $147$ positive divisors, what is the minimum possible value of $n$ ?

Answer: 1080

Answer

The minimum possible value of

n

is 1080.

By prime factorizing

n = p^a q^b r^c

, the number of divisors of

n^2

is given by

(2a+1)(2b+1)(2c+1) = 147

. Decomposing 147 into three factors greater than 1 gives

7 \times 7 \times 3

, which determines the exponents

a=3

b=3

, and

c=1

. Minimizing

n

by assigning the smallest primes (2, 3, and 5) to these exponents yields

2^3 \times 3^3 \times 5^1 = 1080

Step-by-Step Solution

Set up the prime factorization of

n

and

n^2

Let

n = p^a q^b r^c

, where

p < q < r

are prime numbers and

a, b, c \ge 1

. Then

n^2 = p^{2a} q^{2b} r^{2c}

Any positive integer can be uniquely represented as a product of prime powers.

Formulate the divisor counting equation for

n^2

The total number of positive divisors of

n^2

(2a+1)(2b+1)(2c+1) = 147

If an integer has prime factorization

\prod p_i^{e_i}

, its number of positive divisors is

\prod (e_i + 1)

Factor 147 into three odd factors greater than 1

The prime factorization of 147 is

3 \times 7^2

. The only factorization into three integers greater than 1 is

7 \times 7 \times 3

Since

a, b, c \ge 1

, we have

2a+1 \ge 3

2b+1 \ge 3

, and

2c+1 \ge 3

Solve for the exponents

a, b, c

Setting

2a+1=7

2b+1=7

, and

2c+1=3

yields

a=3

b=3

, and

c=1

Equating the factor values allows us to determine the required exponents of the prime factors of

n

Minimize

n

by choosing prime bases and assigning exponents

Choose the three smallest primes

p=2, q=3, r=5

and assign exponents

3, 3, 1

to get

n = 2^3 \times 3^3 \times 5^1 = 1080

To make the product as small as possible, smaller prime bases must be paired with larger exponents.

Key Concept

Determining the prime factorization and number of divisors of prime power expressions

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