### Instruction: What is the smallest positive integer that is a multiple of both 30 and 40 but not a multiple of 16?### Response: I want to find the smallest positive integer that is a multiple of both 30 and 40, so I need to find the least common multiple of 30 and 40.
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.
I can do this by dividing 120 by 16 and seeing if I get a whole number.
120 / 16 = 7.5, which is not a whole number, so 120 is not a multiple of 16.