The least number greater than 16 which, when divided by 4, 6, 8, 12 and 16,…

Concept: When a number greater than the given divisors must leave the same remainder r on division by each of them, the least such number equals the LCM of the divisors plus r — because the LCM is the smallest number exactly divisible by every divisor (remainder 0 in each case), and adding r shifts every division's remainder from 0 up to r without exceeding any divisor (since r is smaller than each one here).

Application:

1. The prime factorisations of the divisors are: 4 = 22, 6 = 2 × 3, 8 = 23, 12 = 22 × 3, 16 = 24.

2. Taking the highest power of each prime that appears -- 24 from 16, and 3 from 6 or 12 -- the LCM = 24 × 3 = 48.

3. Add the required remainder of 2 to this LCM: 48 + 2 = 50.

Cross-check: Dividing 50 by 4, 6, 8, 12 and 16 gives quotients 12, 8, 6, 4 and 3 respectively — each with a remainder of exactly 2, confirming 50 satisfies the condition for every divisor. The only smaller number satisfying all five conditions simultaneously is 2 itself (48k + 2 for k = 0), but 2 is smaller than every divisor and so is not a meaningful case of division with a nonzero quotient; the least number greater than the divisors is 50.