What is the smallest number that leaves a remainder of 2 when divided by 3, 4,…

Key idea: the number must be 2 more than a number divisible by 3, 4, 5, and 6.

1. Rewrite the condition as n ≡ 2 (mod 3, 4, 5, 6), so n - 2 must be a common multiple of 3, 4, 5, and 6.

2. Compute the least common multiple: lcm(3, 4, 5, 6). Use prime powers: 4 contributes 2^2, 3 contributes 3, and 5 contributes 5. So lcm = 2^2 × 3 × 5 = 60.

3. The smallest number of the required form is 2 more than the least common multiple: 60 + 2 = 62.

Check: 62 mod 3 = 2, 62 mod 4 = 2, 62 mod 5 = 2, 62 mod 6 = 2, so 62 satisfies all conditions.