Let d(k) denote the number of positive divisors of a positive integer k. What is the small… — GMAT

Let $d(k)$ denote the number of positive divisors of a positive integer $k$ . What is the smallest positive integer $n$ that is a multiple of $72$ and satisfies $d(n) = 35$ ?

Answer: 5184

Answer

The smallest positive integer

n

that satisfies all conditions is 5184.

The prime factorization of

72

2^3 \times 3^2

. Any multiple

n

72

must be written as

2^a \times 3^b \times dots

with

a \ge 3

and

b \ge 2

. The total number of positive divisors

d(n) = (a+1)(b+1) dots = 35

. The integer

35

can be factored into integers greater than

1

only as

7 \times 5

. Thus,

n

must have exactly two distinct prime factors, which must be

2

and

3

. The required exponents are

a, b \in \{6, 4\}

. To minimize

n

, we assign the larger exponent

6

to the smaller base

2

, yielding

n = 2^6 \times 3^4 = 64 \times 81 = 5184

Step-by-Step Solution

Express 72 in prime factor form to determine minimum required prime exponents.

72 = 2^3 \times 3^2

, meaning

n

must have prime factorization

2^a \times 3^b \dots

where

a \ge 3

and

b \ge 2

Any multiple of 72 must contain at least three factors of 2 and two factors of 3.

Analyze the divisor count condition

d(n) = 35

Since

35 = 7 \times 5

, the number of prime factors of

n

can be at most 2, with exponent increments

(a+1)(b+1) = 7 \times 5

The number of positive divisors for

p_1^{e_1} p_2^{e_2} \dots

(e_1+1)(e_2+1)\dots

, and 35 factors into integers greater than 1 only as 35 or

7 \times 5

Determine the required exponents for the prime factors of

n

The exponents of the prime factors must be

7-1 = 6

and

5-1 = 4

Since

n

must contain both 2 and 3 as prime factors,

n

cannot have only 1 prime factor, so it has exactly two prime factors (2 and 3).

Optimize the exponent assignment to minimize

n

Assign exponent 6 to base 2 and exponent 4 to base 3, giving

n = 2^6 \times 3^4 = 64 \times 81 = 5184

Assigning the larger exponent to the smaller prime base minimizes the total product while meeting the conditions

a \ge 3

and

b \ge 2

Key Concept

Divisor counting formula and prime factor exponent allocation under divisibility constraints

Estimated Time:2m 0s

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