The prime factor decompositions of two numbers are:

6435 = 3² x 5 x 11 x 13
8190 = 2 x 3² x 5 x 7 x 13

Which of the prime factor decompositions below are common factors of 6435 and 8190?

A. 2 x 3 x 5 x 7 x 11 x 13
B. 2 x 3
C. 3 x 13
D. 3² x 5 x 13
E. 3²
F. 2 x 7 x 11
G. 2 x 3⁴ x 5² x 7 x 11 x 13²

By comparing the prime factor decompositions of 6435 and 8190, the highest common factors present in both numbers are 3², 5, and 13. Therefore, option D: 3² × 5 × 13 is the correct answer.

Explanation:

To find the common factors of the numbers 6435 and 8190 based on their prime factor decompositions, we need to identify the prime numbers that appear in the decompositions of both numbers and use the smallest powers of these primes that are present in both decompositions.

The prime factor decomposition of 6435 is 3² × 5 × 11 × 13, and for 8190, it is 2 × 3² × 5 × 7 × 13. Therefore, the common prime factors are 3², 5, and 13. The common factors must all be present in both decompositions. The highest common factor from the options presented that occurs in both numbers is therefore option D: 3² × 5 × 13.