Find the largest number of four digits which gives remainder 4 when divided by…

Answer: 9964

Reason:

1. If a number leaves remainder 4 when divided by 6, 8, 10 and 12, then (number − 4) must be divisible by each of these divisors.

2. Compute the least common multiple: LCM(6, 8, 10, 12) = 120.

3. So the required numbers have the form 120k + 4. The largest four-digit number is 9999, so solve 120k + 4 ≤ 9999.

4. 120k ≤ 9995 ⇒ k ≤ 83.291…, so the largest integer k is 83.

5. Thus the largest such number is 120 × 83 + 4 = 9960 + 4 = 9964.

6. Check: 9964 − 4 = 9960 = 120 × 83, so it is divisible by 6, 8, 10 and 12. Therefore 9964 is correct.