The GCF of 16 32 and 72

The Correct Answer and Explanation is:

Correct Answer:

The greatest common factor (GCF) of 16, 32, and 72 is 8.

Explanation:

The greatest common factor (GCF) of a set of numbers is the largest number that evenly divides all the given numbers. To determine the GCF of 16, 32, and 72, we follow these steps:

Step 1: Prime Factorization

16:
( 16 = 2 \times 2 \times 2 \times 2 = 2^4 )
32:
( 32 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5 )
72:
( 72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2 )

Step 2: Identify Common Factors

The common prime factor for all three numbers is ( 2 ).

The powers of ( 2 ) in each number are:
( 2^4 ) for 16,
( 2^5 ) for 32,
( 2^3 ) for 72.

The smallest power of ( 2 ) common to all numbers is ( 2^3 ).

Step 3: Multiply Common Factors

The product of the common factors gives the GCF:
[ \text{GCF} = 2^3 = 8 ]

Verification

To confirm, divide each number by 8:

( 16 \div 8 = 2 ) (no remainder),
( 32 \div 8 = 4 ) (no remainder),
( 72 \div 8 = 9 ) (no remainder).

Since 8 evenly divides all three numbers, it is the GCF.

Why the GCF is Useful

The GCF is used in simplifying fractions, determining shared factors in problems, and optimizing solutions in real-world applications like dividing items into equal parts. Here, it shows the largest value by which 16, 32, and 72 can be uniformly divided.