What is the greatest common factor (GCF) of the numbers 36, 135, and 144?

To determine the greatest common factor (GCF) of the numbers 36, 135, and 144, it is essential to find the prime factorization of each number and then identify the highest factors they have in common.

Starting with 36, its prime factorization is:

36 = 2 × 2 × 3 × 3 = (2^2 \times 3^2).

Next, for 135, the prime factorization is:

135 = 3 × 3 × 3 × 5 = (3^3 \times 5).

For 144, the prime factorization is:

144 = 2 × 2 × 2 × 2 × 3 × 3 = (2^4 \times 3^2).

Now, to find the GCF, look at the common prime factors of all three numbers. The common prime factor here is 3, and we take the lowest exponent of this common prime factor from the factorizations:

• For 36, the exponent of 3 is 2.

• For 135, the exponent of 3 is 3.

• For 144, the exponent of 3 is 2.

The lowest