What is the total number of positive even factors of the integer 180? — State PSC Exam

What is the total number of positive even factors of the integer $180$ ?

The total number of positive even factors of

180

12

The prime factorization of

180

2^2 \times 3^2 \times 5^1

. An even factor must be divisible by

2

, so the power of

2

must be either

1

2

(

2

choices). The power of

3

can be

0, 1,

2

(

3

choices), and the power of

5

can be

0

1

(

2

choices). Total even factors =

2 \times 3 \times 2 = 12

Express

180

in terms of its prime factors.

180 = 2^2 \times 3^2 \times 5^1

Prime factorization allows systematic calculation of factor counts using exponent combinations.

Determine the number of valid choices for each prime factor's exponent to form an even factor.

Choices for exponent of

2

1

2

(

2

choices). Choices for exponent of

3

0, 1

, or

2

(

3

choices). Choices for exponent of

5

0

1

(

2

choices).

An even factor must contain at least one factor of

2

, so the exponent of

2

cannot be

0

Multiply the counts of choices together.

2 \times 3 \times 2 = 12

By the fundamental principle of counting, multiplying the available choices for each prime factor gives the total number of even factors.

Counting Even Factors using Prime Factorization

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