Find the least non-negative integer that must be added to 1057 so that the result is exactly divisible by 23. What value should be added?

Introduction / Context:This problem tests modular arithmetic and divisibility. We want the smallest non-negative addition to 1057 that makes the sum a multiple of 23. Such questions appear frequently in aptitude tests to assess number sense and comfort with remainders.

Given Data / Assumptions:

• Base number N = 1057.
• Divisor d = 23.
• We seek the least k >= 0 such that (N + k) is divisible by d.

Concept / Approach:If r is the remainder when N is divided by d, then N = q*d + r. The smallest k that makes N + k a multiple of d is k = (d - r) when r != 0, and k = 0 when r = 0. We compute r using N mod d.

Step-by-Step Solution:

Verification / Alternative check:Direct division shows 23 * 46 = 1058; being one less means we must add 1 to reach the next multiple. This confirms k = 1.

Why Other Options Are Wrong:2, 3, 4, and 6 overshoot the next multiple; adding any of them does not land exactly on a multiple of 23.

Common Pitfalls:Confusing remainder 22 with the required addition; remember to add (23 - 22) and not 22 itself.

Final Answer:1