Find the smallest 4-digit number which, when divided by 3, 4, or 5, leaves a remainder of 2 in each case.

Introduction:
This is a modular arithmetic and least common multiple (LCM) question. You are asked to find the smallest 4-digit number which leaves the same remainder (2) when divided by 3, 4, or 5. Such problems test the understanding of congruences and the idea that numbers with the same remainder for given divisors can be written in terms of the LCM of these divisors plus that remainder.

Given Data / Assumptions:

• We want a 4-digit number N (i.e., N ≥ 1000).
• N leaves remainder 2 when divided by 3, 4, and 5.
• So N ≡ 2 (mod 3), N ≡ 2 (mod 4), and N ≡ 2 (mod 5).
• We must find the smallest such N.

Concept / Approach:
If a number leaves the same remainder 2 with respect to 3, 4, and 5, then N - 2 must be divisible by each of these numbers. Therefore, N - 2 must be a multiple of the least common multiple (LCM) of 3, 4, and 5. Once we find the LCM, we can express N as N = LCM(3,4,5) * k + 2 and search for the smallest k that makes N at least 1000.

Step-by-Step Solution:
Compute LCM of 3, 4, and 5.LCM(3,4) = 12; then LCM(12,5) = 60.So N - 2 must be a multiple of 60.Let N - 2 = 60k => N = 60k + 2.We need N to be at least 1000.So 60k + 2 ≥ 1000 => 60k ≥ 998 => k ≥ 998 / 60 ≈ 16.63.The smallest integer k satisfying this is k = 17.Then N = 60 * 17 + 2 = 1020 + 2 = 1022.

Verification / Alternative check:
Check N = 1022: 1022 / 3 leaves remainder 2; 3 * 340 = 1020, remainder 2. 1022 / 4 leaves remainder 2; 4 * 255 = 1020, remainder 2. 1022 / 5 leaves remainder 2; 5 * 204 = 1020, remainder 2. Thus 1022 satisfies all three conditions and is the smallest 4-digit number of the form 60k + 2.

Why Other Options Are Wrong:
1020, 1040, 1060, and 1122: these are not of the form 60k + 2 or do not leave remainder 2 with all three divisors 3, 4, and 5. For example, 1020 is exactly divisible by 3, 4, and 5 with remainder 0, not 2.

Common Pitfalls:
Trying to test random 4-digit numbers instead of using the LCM structure.Forgetting that N - 2 must be divisible by all three numbers, not N itself.Rounding 998/60 down instead of up when selecting k, which would produce a 3-digit or smaller number.

Final Answer:
1022