What is the greatest common factor of 36 and 54? — ACT

What is the greatest common factor of $36$ and $54$ ?

Answer: 18

Answer

The greatest common factor of

36

and

54

18

The greatest common factor of

36

and

54

18

because it is the largest integer that divides both numbers evenly (

36 \div 18 = 2

and

54 \div 18 = 3

Step-by-Step Solution

Find the prime factorization of

36

and

54

36 = 2^2 \times 3^2

and

54 = 2 \times 3^3

Prime factorization decomposes each number into its basic prime building blocks.

Identify the common prime factors with their lowest exponents.

The common prime factors are

2

with an exponent of

1

(

2^1

) and

3

with an exponent of

2

(

3^2

The greatest common factor consists of the product of the lowest powers of the shared prime factors.

Multiply the common factors raised to their lowest exponents.

2^1 \times 3^2 = 2 \times 9 = 18

Calculating this product gives the greatest common factor.

Key Concept

The greatest common factor (GCF) of two integers is the largest integer that divides both numbers without leaving a remainder. It is calculated by taking the product of the lowest powers of all common prime factors.

Alternative Method

List all factors of both numbers: Factors of

36

are

1, 2, 3, 4, 6, 9, 12, 18, 36

. Factors of

54

are

1, 2, 3, 6, 9, 18, 27, 54

. The largest number present in both lists is

18

Estimated Time:45s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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