Correct Option (B)
To determine the least four-digit number that leaves a remainder of 2 when divided by 3, 4, 5, and 6, the following steps are applied:
- First, calculate the Least Common Multiple (LCM) of the divisors: 3, 4, 5, and 6.
- Prime factorization: 3 = 3, 4 = 2², 5 = 5, 6 = 2 × 3.
- LCM = 2² × 3 × 5 = 4 × 3 × 5 = 60.
- Numbers that leave a remainder of 2 when divided by 3, 4, 5, and 6 are of the form (60k + 2), where 'k' is a positive integer.
- Identify the least four-digit number. This is 1000.
- Find the smallest multiple of 60 that is greater than or equal to 1000.
- Divide 1000 by 60: 1000 ÷ 60 = 16 with a remainder of 40.
- This indicates that 1000 is 40 more than 960 (60 × 16).
- The next multiple of 60 is 960 + 60 = 1020. This is the least four-digit number exactly divisible by 3, 4, 5, and 6.
- Add the required remainder (2) to this number: 1020 + 2 = 1022.
- Therefore, 1022 is the least four-digit number satisfying the given conditions.
Incorrect Options:
Options A, C, and D are incorrect because they do not satisfy the specified conditions:
- For a number to leave a remainder of 2 when divided by 3, 4, 5, and 6, the number minus 2 must be perfectly divisible by their LCM, which is 60.
- Option (A) 1012: 1012 - 2 = 1010. 1010 is not divisible by 60 (1010 ÷ 60 = 16 with remainder 50).
- Option (C) 1122: 1122 - 2 = 1120. 1120 is not divisible by 60 (1120 ÷ 60 = 18 with remainder 40).
- Option (D) 1222: 1222 - 2 = 1220. 1220 is not divisible by 60 (1220 ÷ 60 = 20 with remainder 20).
- Furthermore, even if these options satisfied the divisibility condition, they would not represent the least four-digit number, as 1022 is smaller.