To determine which perfect squares are factors of 72, let's first find the prime factorization of 72.
The prime factorization of 72 is: \[ 72 = 2^3 \times 3^2 \]
Now, we will examine each of the perfect squares listed to see if they are factors of 72:
-
36:
- \( 36 = 6^2 = 2^2 \times 3^2 \)
- Since \( 36 \) can be made from the primes in \( 72 \), it is a factor of \( 72 \).
-
16:
- \( 16 = 4^2 = 2^4 \)
- \( 16 \) cannot be formed from the prime factors of \( 72 \) because \( 72 \) has at most \( 2^3 \). Thus, \( 16 \) is not a factor of \( 72 \).
-
25:
- \( 25 = 5^2 \)
- \( 25 \) does not contain any of the prime factors of \( 72 \). Thus, \( 25 \) is not a factor of \( 72 \).
-
4:
- \( 4 = 2^2 \)
- Since \( 4 \) can be made from the primes in \( 72 \), it is a factor of \( 72 \).
-
9:
- \( 9 = 3^2 \)
- Since \( 9 \) can be made from the primes in \( 72 \), it is a factor of \( 72 \).
So, the perfect squares that are factors of 72 are:
- 36
- 4
- 9