What is the greatest integer less than 100 for which the greatest common divisor of that integer and 12 is 4?
I know that the gcd of two numbers is the largest factor that they both share, so n must be a multiple of 4. The largest multiple of 4 that is less than 100 is 96, but gcd(96, 12) = 12, not 4. So I need to go down to the next smaller multiple of 4, which is 92. Is gcd(92, 12) = 4? I can use the Euclidean algorithm to check: 92 = 12 * 7 + 8, 12 = 8 * 1 + 4, 8 = 4 * 2 + 0. The last nonzero remainder is 4, so gcd(92, 12) = 4. Therefore, 92 is the largest number n that satisfies the conditions. 