What are the numbers whose triples, when added with 1, result in a prime number between 70 and 110?
The prime numbers between 70 and 110 are

71, 73, 79, 83, 89, 97, 101, 103, 107, and 109.

Subtracting 1 from all these numbers, we get the list

$$
70, \quad 72, \quad 78, \quad 82, \quad 88, \quad 96, \quad 100, \quad 102, \quad 106 \text { and } 108
$$

From this list, the multiples of 3 are 72, 78, 96, 102, and 108. Therefore, the numbers we are looking for are

$$
24=72 \div 3,26=78 \div 3,32=96 \div 3,34=102 \div 3 \text { and } 36=108 \div 3
$$

Indeed, we have $24 \times 3+1=73,26 \times 3+1=79,32 \times 3+1=97,34 \times 3+1=103 \text{ and } 36 \times 3+1=109$.