What is the least number that must be added to 1039 so that the resulting sum is exactly divisible by 29?

Introduction / Context:
This question tests your understanding of divisibility and remainders. You must find the smallest number that can be added to a given integer so that the new total becomes perfectly divisible by a specified divisor. Such problems are common in quantitative aptitude, number system, and basic arithmetic sections of competitive exams.

Given Data / Assumptions:
- The starting number is 1039.
- The divisor is 29.
- We want the smallest non negative integer that can be added to 1039 so that the resulting sum is divisible by 29 without any remainder.
- Division is in the usual integer sense.

Concept / Approach:
The key idea is to use the concept of remainder. When a number N is divided by a divisor d, we can write N = d * q + r, where q is the quotient and r is the remainder. To make N divisible by d, we need to add a value that cancels out the remainder and lifts the total up to the next multiple of d. The least required addition is d - r, unless the remainder is zero already.

Step-by-Step Solution:
Step 1: Divide 1039 by 29 to find the remainder.Step 2: Compute 29 * 35 = 1015, which is the largest multiple of 29 less than or equal to 1039.Step 3: Find the remainder: 1039 - 1015 = 24.Step 4: This means 1039 = 29 * 35 + 24.Step 5: To reach the next multiple of 29, we need to add an amount that makes the total remainder zero.Step 6: The next multiple above 1039 is 29 * 36 = 1044.Step 7: The difference between 1044 and 1039 is 1044 - 1039 = 5.Step 8: Therefore, the least number to be added to 1039 to make it exactly divisible by 29 is 5.

Verification / Alternative check:
After adding 5, the new number is 1039 + 5 = 1044. Now divide 1044 by 29. We know that 29 * 36 = 1044, so the quotient is 36 and the remainder is 0. This confirms that 1044 is perfectly divisible by 29 and that adding 5 is sufficient and minimal because any smaller addition would leave a nonzero remainder.

Why Other Options Are Wrong:
If you add 4, the sum becomes 1043, which gives a remainder of 28 when divided by 29. Adding 6 gives 1045, which is larger than necessary, and 1045 divided by 29 still gives a remainder. Values like 8 or 9 move even farther away from the next exact multiple and do not represent the least required addition. Only adding 5 produces the next exact multiple of 29.

Common Pitfalls:
Learners sometimes mistakenly divide the divisor by the remainder or try to guess the answer rather than using the remainder method. Another frequent mistake is calculating the wrong multiple of the divisor due to multiplication errors. Carefully finding the largest multiple of 29 below 1039 and then computing the difference gives a reliable answer every time.

Final Answer:
The least number that must be added is 5.