5. A three-digit number that is divisible by 35 and whose digits sum to 15 is
5. 735.

Since the sum of the digits is 15, the three-digit number is a multiple of 3. Also, since the three-digit number is a multiple of 35, this three-digit number is a multiple of $35 \times 3=105$.
Starting from 105, list the three-digit multiples of 105:
$$
105,210,315,420,525,630,735,840, 945 \text {, }
$$

Among them, only 735 has a digit sum of 15.