Which smallest number must be subtracted from 400 so that the resulting number is exactly divisible by 7?

Introduction / Context:
This is a divisibility and remainder problem. The question asks by how much we need to decrease 400 so that the new number is divisible by 7. It checks understanding of remainders and how they relate to adjusting a number to become a multiple of a divisor.

Given Data / Assumptions:

- Original number = 400.

- We subtract a smallest positive integer k from 400.

- The resulting number 400 - k must be divisible by 7.

Concept / Approach:
If a number leaves a remainder r when divided by 7, then subtracting r from that number will make it divisible by 7. So we first find 400 modulo 7, that is the remainder when 400 is divided by 7. That remainder itself will be the smallest non negative number to subtract.

Step-by-Step Solution:
Step 1: Divide 400 by 7 to find the remainder. 7 × 57 = 399. Subtract: 400 - 399 = 1. So the remainder when 400 is divided by 7 is 1. Step 2: Use the remainder to find the required subtraction. If 400 leaves remainder 1, then 400 - 1 = 399. Step 3: Check divisibility of 399 by 7. 399 / 7 = 57 exactly, so 399 is divisible by 7. Therefore, the smallest number to be subtracted from 400 is 1.

Verification / Alternative check:
Try other small subtractions to confirm: 400 - 2 = 398. 398 / 7 = 56 remainder 6, not divisible. 400 - 4 = 396. 396 / 7 = 56 remainder 4, not divisible. So none of the larger subtractions gives a multiple of 7 before subtracting 1, confirming that 1 is indeed the least.

Why Other Options Are Wrong:
- 6: 400 - 6 = 394, which is not divisible by 7.
- 2: 398 is not divisible by 7.
- 4: 396 is still not divisible by 7.

Common Pitfalls:
Some students attempt to divide each candidate (400 - option) by 7 directly, which works but is slower. Others confuse “subtract” with “add” and mistakenly look for a number to add to 400. Remember that the remainder gives an immediate clue to how much you need to subtract to reach the nearest lower multiple of 7.

Final Answer:
The smallest number that must be subtracted from 400 is 1.