A positive integer K is a multiple of 10 and has exactly 8 positive factors (including 1 a… — ACT

A positive integer $K$ is a multiple of $10$ and has exactly $8$ positive factors (including $1$ and $K$ ). What is the least possible value of $K$ ?

Cevap: 30

The least possible value of

K

30

The correct answer is

30

because it is the smallest multiple of

10

that has exactly

8

positive factors (

1, 2, 3, 5, 6, 10, 15,

and

30

Determine the prime factors required for a multiple of

10

K

must have at least one factor of

2

and at least one factor of

5

Since

10 = 2 \times 5

, any multiple of

10

must be divisible by both

2

and

5

Set up the factor counting formula.

The number of factors is

(a+1)(b+1)(c+1)\cdots = 8

, where

a, b, c, \dots

are the exponents of the prime factorization.

The number of positive divisors of

p_1^a p_2^b \cdots

is found by adding

1

to each exponent and multiplying the results.

Evaluate the smallest multiples of

10

to find the first one with exactly

8

factors.

10

has

4

factors (

1, 2, 5, 10

);

20

has

6

factors (

1, 2, 4, 5, 10, 20

);

30

has

8

factors (

1, 2, 3, 5, 6, 10, 15, 30

Testing the positive multiples of

10

in ascending order ensures that the first number with exactly

8

factors is the smallest possible value.

Factors, Multiples, and Prime Factorization

Soru