1. [3] Find the number of integers between 1 and 200 inclusive whose distinct prime divisors sum to 16 . (For example, the sum of the distinct prime divisors of 12 is $2+3=5$.)
Answer: 6 The primes less than 16 are $2,3,5,7,11$, and 13 . We can write 16 as the sum of such primes in three different ways and find the integers less than 200 with those prime factors:
- $13+3: 3 \cdot 13=39$ and $3^{2} \cdot 13=117$.
- $11+5: 5 \cdot 11=55$
- $11+3+2: 2 \cdot 3 \cdot 11=66,2^{2} \cdot 3 \cdot 11=132$, and $2 \cdot 3^{2} \cdot 11=198$.

There are therefore 6 numbers less than 200 whose prime divisors sum to 16 .