Question
What is the greatest four-digit number that leaves
remainders 1, 2 and 3 respectively when divided by 2, 3 and 4?Solution
ATQ, Let the required number be N. Given: N ≡ 1 (mod 2) N ≡ 2 (mod 3) N ≡ 3 (mod 4) Note that these remainders are each one less than the divisor: N ≡ −1 (mod 2), (mod 3), (mod 4) ⇒ N + 1 is divisible by 2, 3 and 4. So N + 1 is a multiple of LCM(2,3,4) = 12. Let N + 1 = 12k. For largest 4-digit N: N ≤ 9999 ⇒ 12k − 1 ≤ 9999 ⇒ 12k ≤ 10000 ⇒ k ≤ 833. Max k = 833 ⇒ N = 12 × 833 − 1 = 9996 − 1 = 9995.
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