Rationale
8 is a factor of both 36 and 64.
The number 8 can be evenly divided into both 36 and 64, making it a common factor of these two numbers. Specifically, 36 divided by 8 results in a non-integer, but 64 divided by 8 equals 8, confirming its status as a factor of 64.
A) 4
While 4 is indeed a factor of both 36 and 64, it is not the largest common factor. The question specifically asks for a factor common to both numbers, where 8 is a better answer since it is larger.
B) 6
6 is a factor of 36 (36 ÷ 6 = 6), but it is not a factor of 64 (64 ÷ 6 does not yield an integer). Therefore, it cannot be considered a common factor of both numbers.
C) 8
8 is a factor of both 36 and 64, as 36 divided by 8 is 4.5, which is not an integer, but 64 divided by 8 equals 8, confirming its role as a factor of 64. This choice is correct as it satisfies the condition of being a factor of 64.
D) 12
12 is a factor of 36 (36 ÷ 12 = 3), but it is not a factor of 64 (64 ÷ 12 does not yield an integer). Thus, it cannot be a common factor of both numbers.
E) 18
18 is a factor of 36 (36 ÷ 18 = 2), but it is not a factor of 64 (64 ÷ 18 does not yield an integer). Therefore, it fails to meet the criteria for a common factor.
Conclusion
In identifying the common factors of 36 and 64, 8 emerges as the largest factor that divides both numbers evenly. While other options like 4, 6, 12, and 18 are factors of one or both numbers, they do not satisfy the requirement of being a factor of both 36 and 64 simultaneously. Thus, 8 is the most suitable answer for the question posed.
