Find the least number which, when divided by 8, 12 and 15, leaves remainders 5, 9 and 12 respectively.

Explanation

1. 01

   Divisor − remainder is the same each time: $8 - 5 = 12 - 9 = 15 - 12 = 3$.

2. 02

   So the number is 3 less than a common multiple of 8, 12 and 15.

3. 03

   $\operatorname{LCM}(8, 12, 15) = 120$, so the least number is $120 - 3 = 117$.

4. 04

   Check: $117 = 8 \times 14 + 5 = 12 \times 9 + 9 = 15 \times 7 + 12$.