The prime factorizations of two integers, X and Y, are given below: X = 2^3 \times 3^2 \ti… — Armed Services Vocational Aptitude Battery (ASVAB)

The prime factorizations of two integers, $X$ and $Y$ , are given below:

$X = 2^3 \times 3^2 \times 5$
$Y = 2^2 \times 3^3$

Based on this information, what is the greatest common factor (GCF) of $X$ and $Y$ ?

$36$ Answer
B
$180$
C
$1080$
D
$6$

Answer

The greatest common factor is

36

To determine the greatest common factor (GCF) from prime factorizations, we must identify the prime bases shared by both numbers and raise each to its lowest given exponent. The shared bases for

X

and

Y

are

2

and

3

. Taking the lowest exponent for each gives

2^2

and

3^2

. Their product,

4 \times 9

, equals

36

Step-by-Step Solution

Identify the common prime bases in both factorizations.

The prime bases present in both

X

and

Y

are

2

and

3

. The prime base

5

is only in

X

The greatest common factor (GCF) only includes prime factors that are shared by all given numbers.

Determine the lowest exponent for each common prime base.

For base

2

, the exponents are

3

and

2

, so the lowest is

2

. For base

3

, the exponents are

2

and

3

, so the lowest is

2

To find the greatest common factor, we must use the smallest available power of each common prime to ensure it divides both numbers perfectly.

Multiply the resulting prime powers to calculate the GCF.

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

The product of these prime powers yields the greatest integer that divides both

X

and

Y

without a remainder.

Key Concept

Calculating the greatest common factor (GCF) using prime factorization by finding the lowest powers of all common prime bases.

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