Question

If gcd(a, b) = 1 and gcd(a,c) = 1, then what is the gcd(a, bc)?

1) 1
2) a
3) b
4) c

Answer 1

The gcd of 'a' and 'bc' is 1 since 'a' has no common prime factors with 'b' or 'c', and multiplying 'b' and 'c' does not introduce any new common factors with 'a'.

Step-by-step explanation:

The greatest common divisor (gcd) of two integers is the largest positive integer that divides both of them without leaving a remainder. Given that gcd(a, b) = 1 and gcd(a,c) = 1, it means that 'a' shares no common prime factors with 'b' or 'c'.

To find the gcd(a, bc), we can use the property that if gcd(a, b) = 1, then for any integer 'c', gcd(a, bc) will also be 1. This is because 'a' does not have any common factors with 'b', and multiplying 'b' by 'c' does not introduce any new common factors with 'a'.

Hence, gcd(a, bc) = 1 even if 'b' and 'c' are multiplied together. The correct answer to the question "What is the gcd(a, bc)?" is therefore option 1).