What is the smallest positive integer that is a multiple of both 30 and 40 but not a multiple of 16?
One way to do this is to write the prime factorization of both numbers and take the highest power of each prime factor. 30 = 2 * 3 * 5 and 40 = 2^3 * 5, so the least common multiple is 2^3 * 3 * 5 = 120. Now I need to check if 120 is a multiple of 16. 120 / 16 = 7.5, which is not a whole number, so 120 is not a multiple of 16. Therefore, 120 is the smallest positive integer that is a multiple of both 30 and 40 but not a multiple of 16.

