What is the smallest number that must be subtracted from 518 so that the resulting number is exactly divisible by 13?

Introduction / Context:
This question is about divisibility and remainders. We are asked to find the least number that must be subtracted from 518 so that the result is divisible by 13. This tests understanding of how remainders behave when adjusting a number by subtraction.

Given Data / Assumptions:
Original number is 518.We seek an integer k such that 518 − k is divisible by 13.Among the options, we must choose the smallest such k.

Concept / Approach:
When a number N is divided by a divisor d, we can write N = dq + r, where r is the remainder. To get the nearest smaller multiple of d, we must subtract the remainder r from N. Therefore, the smallest non negative number that must be subtracted is exactly the remainder when N is divided by d.

Step-by-Step Solution:
Step 1: Divide 518 by 13 to find the remainder.Step 2: Compute 13 × 39 = 507.Step 3: Subtract 507 from 518: 518 − 507 = 11.Step 4: This shows that 518 = 13 × 39 + 11, so the remainder when 518 is divided by 13 is 11.Step 5: To get the largest multiple of 13 that is less than or equal to 518, subtract the remainder from 518.Step 6: So the required multiple is 518 − 11 = 507, which is already known to be divisible by 13.Step 7: Therefore, the least number that must be subtracted is 11.

Verification / Alternative check:
Check divisibility: 507 ÷ 13 = 39 exactly, with no remainder. If we were to subtract a smaller number, say 10, we would get 508, and 508 ÷ 13 is not an integer. Similarly, subtracting 9 gives 509, which is also not divisible by 13. Therefore, 11 is the smallest subtraction that makes the number divisible by 13.

Why Other Options Are Wrong:
Subtracting 10 yields 508, and 13 × 39 = 507 and 13 × 40 = 520, so 508 is not a multiple of 13. Subtracting 9 or 12 gives results with remainders 2 or 10 when divided by 13. These are not exact multiples. Only subtracting 11 eliminates the remainder completely.

Common Pitfalls:
Some learners mistakenly divide in the wrong direction or try random subtractions from the options. Others may miscalculate the product 13 × 39 or forget that the remainder itself is the quantity that must be removed. Always compute the remainder first, then subtract exactly that amount to reach the nearest smaller multiple.

Final Answer:
The smallest number that must be subtracted is 11.