What is the smallest integer greater than 200 for which the greatest common divisor of that integer and 36 is 18?
That means n must be a multiple of 18, since 18 is the largest factor that both n and 36 share. The smallest multiple of 18 that is greater than 200 is 216, which is 18 times 12. But 216 is also a multiple of 36, so the gcd of 216 and 36 is 36, not 18. I need to find the next multiple of 18 that is not a multiple of 36. That would be 234, which is 18 times 13. To check, I can use the Euclidean algorithm to find the gcd of 234 and 36. I start by dividing 234 by 36 and getting a quotient of 6 and a remainder of 18. Then I divide 36 by 18 and get a quotient of 2 and a remainder of 0. That means the gcd of 234 and 36 is the last nonzero remainder, which is 18. So 234 is the smallest integer greater than 200 that satisfies the given condition. 