What is the total number of positive factors of the integer 36? — GMAT

What is the total number of positive factors of the integer $36$ ?

Answer: 9

The total number of positive factors of

36

is 9.

The prime factorization of

36

2^2 \times 3^2

. By the total factors formula, adding 1 to each exponent and multiplying gives

(2+1)(2+1) = 9

. The factors are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

Find the prime factorization of 36

36 = 4 \times 9 = 2^2 \times 3^2

Decomposing the number into prime factors allows the application of the factor count formula.

Apply the exponent formula for counting factors

(2 + 1)(2 + 1) = 3 \times 3 = 9

Each factor of

36

is of the form

2^x \times 3^y

, where

x \in \{0, 1, 2\}

(3 choices) and

y \in \{0, 1, 2\}

(3 choices).

Calculating the total number of positive factors from prime factorization exponents.

Estimated Time:45s

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