Find the least number which leaves a remainder of 3 when divided by 5, 6, 7, and 8, but is divisible by 9.

Concept / Approach

• If a number leaves the same remainder 3 on division by 5, 6, 7, and 8, then it is of the form N = LCM(5,6,7,8) × k + 3.
• Additionally, N must be divisible by 9.

Step-by-step calculationLCM(5,6,7,8) = 840N = 840k + 3Divisibility by 9: 840k + 3 ≡ 0 (mod 9)840 ≡ 3 (mod 9) ⇒ 3k + 3 ≡ 0 ⇒ 3(k + 1) ≡ 0 (mod 9)k + 1 ≡ 0 (mod 3) ⇒ k ≡ 2 (mod 3)Smallest k = 2 ⇒ N = 840×2 + 3 = 1683

Verification1683 ÷ 9 = 187 (exact); 1683 leaves remainder 3 when divided by 5, 6, 7, and 8 since 1680 is a common multiple.

Final Answer1683