### Instruction: What is the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu,20]=60$?### Response: So we're looking for the least common multiple of 20 and some number v, and that least common multiple is equal to 60.
Right. So let's list out the factors of 20.
Ok. 1, 2, 4, 5, 10, and 20.
Let's also list out the factors of 60.
Ok. 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.
Now let's look for the numbers that are factors of both 20 and 60.
Ok. We see that 1, 2, 4, 5, and 10 are factors of both 20 and 60.
Right. So let's add up all of those numbers.
1 + 2 + 4 + 5 + 10 = 22.