How many numbers between 1 and 400 are divisible by 6?

Concept: To count how many multiples of a number d lie between 1 and N, list them as an arithmetic progression d, 2d, 3d, ... up to the largest multiple that does not exceed N. The count equals the term-position n found from the AP's nth-term formula L = a + (n − 1) × d, where a is the first term and L is the last (largest) term.

Application:

1. Multiples of 6 up to 400 form the sequence 6, 12, 18, ..., with first term a = 6 and common difference d = 6.

2. The largest multiple of 6 not exceeding 400 is L = 396, since 6 × 66 = 396 ≤ 400 while 6 × 67 = 402 > 400.

3. Apply the nth-term formula: L = a + (n − 1) × d, so 396 = 6 + (n − 1) × 6.

4. Solve: 396 − 6 = (n − 1) × 6 ⇒ 390 = (n − 1) × 6 ⇒ n − 1 = 65 ⇒ n = 66.

Cross-check: 400 / 6 = 66.67, and taking the integer part gives 66, matching the AP result independently.

Answer: There are 66 numbers between 1 and 400 that are divisible by 6.