N is a 3-digit number that is a multiple of 7; what is the probability that it…

The classical probability formula gives the chance of an event as the ratio of favorable outcomes to the total number of equally likely outcomes in the sample space.

1. Total outcomes: three-digit multiples of 7 range from 105 (= 15 × 7) to 994 (= 142 × 7), so there are 142 − 15 + 1 = 128 such numbers.

2. Favorable outcomes: N must also be a multiple of 5, so N must be a multiple of the LCM of 7 and 5, which is 35. Three-digit multiples of 35 range from 105 (= 3 × 35) to 980 (= 28 × 35), so there are 28 − 3 + 1 = 26 such numbers.

3. Applying the formula: probability = 26/128 = 13/64.

Cross-check: 15 × 7 = 105 and 142 × 7 = 994 are indeed the smallest and largest three-digit multiples of 7 (143 × 7 = 1001 has four digits). Similarly 3 × 35 = 105 and 28 × 35 = 980 are the smallest and largest three-digit multiples of 35 (29 × 35 = 1015 has four digits). So 26/128 simplifies correctly to 13/64.