How many natural numbers between 1 and 500 are divisible by each of the…

Concept: A number that is divisible by two or more given numbers at the same time must be a multiple of their Least Common Multiple (LCM). So, to count numbers satisfying several divisibility conditions together within a range, first find the LCM of those numbers, then count its multiples that fall within the range.

1. Since 3, 5, and 7 are all prime, they share no common factors, so LCM(3, 5, 7) = 3 × 5 × 7 = 105.

2. A natural number divisible by 3, 5, and 7 together must be a multiple of 105. Multiples of 105 have the form 105k, where k = 1, 2, 3, …

3. We need 105k to lie between 1 and 500, i.e. 105k ≤ 500, so k ≤ 500/105 ≈ 4.76. Since k must be a whole number, k can be 1, 2, 3, or 4.

4. This gives the numbers 105, 210, 315, and 420 — four numbers in total.

Cross-check: The next multiple, 105 × 5 = 525, exceeds 500, confirming no fifth number qualifies. Each of 105, 210, 315, and 420 can also be verified individually — for instance, 420 ÷ 3 = 140, 420 ÷ 5 = 84, and 420 ÷ 7 = 60, all exact.

Hence, four natural numbers between 1 and 500 are divisible by each of 3, 5, and 7.