GCF of 45 and 60

Find the Greatest Common Factor (GCF) of 45 and 60 with a complete step-by-step solution. The GCF is the largest number that divides both numbers without leaving a remainder.

Answer: GCF(45, 60) = 15

Step-by-Step Solution

Step 1 Find the Factors of 45

To find the GCF of 45 and 60, we first need to find all factors of each number.

The factors of 45 are numbers that divide 45 evenly:

1 × 45 = 45, 3 × 15 = 45, 5 × 9 = 45

Factors of 45: 1, 3, 5, 9, 15, 45

Step 2 Find the Factors of 60

Now we find all factors of 60:

1 × 60 = 60, 2 × 30 = 60, 3 × 20 = 60, 4 × 15 = 60, 5 × 12 = 60, 6 × 10 = 60

Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Step 3 Find the Common Factors

The common factors are numbers that appear in both lists:

Factors of 45: 1, 3, 5, 9, 15, 45

Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Common factors: 1, 3, 5, 15

The Greatest Common Factor is the largest of these: 15

Step 4 Prime Factorization Method

We can also find the GCF using prime factorization:

45 = 3^2 × 5

60 = 2^2 × 3 × 5

Take the common prime factors with the lowest powers:

GCF = 3 × 5 = 15

Step 5 Euclidean Algorithm

The Euclidean algorithm is a fast way to find the GCF using repeated division:

60 = 1 × 45 + 15

45 = 3 × 15 + 0

When the remainder is 0, the GCF is the last divisor: 15

Step 6 Final Answer

GCF(45, 60) = 15

GCF of 45 and 60

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