The problem asks to find the greatest common factor (G.C.F.) of the numbers 30, 45, and 60, given their prime factorizations. $30 = 2 \times 3 \times 5$ $45 = 3 \times 3 \times 5$ $60 = 2 \times 2 \times 3 \times 5$

1. Problem Description

The problem asks to find the greatest common factor (G.C.F.) of the numbers 30, 45, and 60, given their prime factorizations.

$30 = 2 \times 3 \times 5$

$45 = 3 \times 3 \times 5$

$60 = 2 \times 2 \times 3 \times 5$

2. Solution Steps

To find the G.C.F., we need to identify the common prime factors in the prime factorizations of the given numbers, and take the lowest power of each.

The prime factors of 30 are 2, 3, and

5. The prime factors of 45 are 3 and

5. The prime factors of 60 are 2, 3, and

5.

The common prime factors of 30, 45, and 60 are 3 and

5.

The lowest power of 3 present in the factorizations is $3^1$.

The lowest power of 5 present in the factorizations is $5^1$.

Therefore, the G.C.F. is the product of these common prime factors raised to their lowest powers.

G.C.F. = $3 \times 5 = 15$

3. Final Answer

15