Correct Option

To determine the smallest number greater than 1000 that leaves a remainder of 3 when divided by 6, 9, 12, 15, and 18, the following steps are applied:

Calculate the Least Common Multiple (LCM) of the divisors:
- Prime factorization of the given numbers:
  - 6 = 2 × 3
  - 9 = 3²
  - 12 = 2² × 3
  - 15 = 3 × 5
  - 18 = 2 × 3²
- The LCM is found by taking the highest power of all prime factors present: LCM(6, 9, 12, 15, 18) = 2² × 3² × 5 = 4 × 9 × 5 = 180.
Formulate the general expression for the number:
A number 'N' that leaves a remainder 'r' when divided by a set of numbers is expressed as `N = k × LCM + r`. In this case, `N = 180k + 3`, where 'k' is a positive integer.
Identify the smallest number greater than 1000:
We need to find the smallest integer 'k' such that `180k + 3 > 1000`.
- `180k > 1000 - 3`
- `180k > 997`
- `k > 997 / 180`
- `k > 5.53...`
The smallest integer value for 'k' that satisfies this condition is 6.
Calculate the specific number:
Substitute k = 6 into the expression: `N = 180 × 6 + 3 = 1080 + 3 = 1083`.

Thus, 1083 is the smallest number greater than 1000 that satisfies the given conditions. When 3 is subtracted from 1083, the result is `1083 - 3 = 1080`, which is perfectly divisible by 180.

Incorrect Options

For any number to leave a remainder of 3 when divided by 6, 9, 12, 15, and 18, subtracting 3 from that number must result in a multiple of their LCM, which is 180.

Option (1) `1063`: `1063 - 3 = 1060`. `1060` is not a multiple of 180 (`1060 ÷ 180 ≈ 5.89`).
Option (2) `1073`: `1073 - 3 = 1070`. `1070` is not a multiple of 180 (`1070 ÷ 180 ≈ 5.94`).
Option (4) `1183`: `1183 - 3 = 1180`. `1180` is not a multiple of 180 (`1180 ÷ 180 ≈ 6.56`).

Therefore, these options do not fulfill the required condition.