A positive integer is said to have exactly 8 positive factors. What is the fifth smallest … — ACT

A positive integer is said to have exactly $8$ positive factors. What is the fifth smallest positive integer that satisfies this condition?

Cevap: 54

The fifth smallest positive integer with exactly 8 positive factors is 54.

The fifth smallest positive integer with exactly 8 positive factors is 54. The number of factors of

n = p_1^{a_1} \cdots p_k^{a_k}

(a_1+1)\cdots(a_k+1) = 8

. The possible factorization shapes are

p^7

(smallest is 128),

p^3 q

(smallest are 24, 40, 54, 56, 88), and

p q r

(smallest are 30, 42, 66, 70). Sorting these gives 24, 30, 40, 42, 54, 56, 66, 70, 78, 88, where 54 is the fifth value.

Relate the number of positive factors to the prime factorization of a positive integer.

The number of positive factors is

(a_1 + 1)(a_2 + 1) \cdots (a_k + 1) = 8

This formula counts all possible combinations of prime factors that form divisors.

Determine the possible prime factorization structures that yield exactly 8 factors.

The possible prime factor structures are

p^7

p^3 q

, and

p q r

for distinct primes

p

q

, and

r

These correspond to the integer factorizations of 8: 8,

4 \times 2

, and

2 \times 2 \times 2

List the smallest candidate values for each structure.

Candidates include 128 (for

p^7

); 24, 40, 54, 56, 88 (for

p^3 q

); and 30, 42, 66, 70 (for

p q r

Evaluating structures with the smallest available prime numbers (2, 3, 5, 7, etc.) yields the smallest positive integers.

Sort all the candidates in ascending order.

The sorted list of smallest values is 24, 30, 40, 42, 54, 56, 66, 70, 78, 88.

Sorting allows us to identify the fifth smallest value precisely.

Identify the fifth value in the sorted list.

The fifth element is 54.

Counting from the smallest value (24) to the fifth position yields 54.

Determining the number of factors of an integer from its prime factorization.

Soru