Question

Suppose b is any integer. If b mod 12 = 7, what is 6b mod 12? In other words, if division of b by 12 gives a remainder of 7, what is the remainder when 6b is divided by 12?

Answer 1

If b mod 12 equals 7, then 6b mod 12 equals 6. This is found by first multiplying the remainder (7) by 6, giving 42, and then finding the remainder when 42 is divided by 12, which is 6.

Step-by-step explanation:

The given question involves finding the modulus of a product of an integer and a constant. Specifically, if b mod 12 = 7, we want to determine what 6b mod 12 is. We approach this by recognizing that multiplying an integer by a constant and then taking a mod does not change the mod of the original integer by a multiple of the modulos, if that multiple itself is modulo the divisor. Since 6 is a multiple of 12, the question reduces to finding what 6 times the remainder of b (which is 7) modulo 12 is.

To solve this, we calculate 6 * 7 = 42. Then, we find 42 mod 12. Since 42 is divisible by 12 with a remainder of 6, our answer is 6b mod 12 = 6.