What is the least number that must be added to 4131 so that the resulting sum is completely divisible by 19?

Introduction / Context:
This question is based on basic divisibility and remainders. The aim is to find the smallest nonnegative integer that should be added to a given number so that the result becomes exactly divisible by a specified divisor. Such problems are very common in aptitude tests because they test understanding of modular arithmetic in a very simple and practical way.

Given Data / Assumptions:

• Given number: 4131.
• Divisor: 19.
• We need to find the least integer k such that 4131 + k is divisible by 19.
• We assume standard properties of division and remainders.

Concept / Approach:
If a number N leaves a remainder r when divided by a divisor d, then N can be written as N = d * q + r. To make the number divisible by d, we need to add a number k so that the new remainder becomes zero. The smallest such k is simply d - r if r is not zero, or 0 if the number is already divisible. So the strategy is to first compute the remainder when 4131 is divided by 19 and then find k = 19 - remainder.

Step-by-Step Solution:
Let N = 4131 and d = 19. Compute the remainder when 4131 is divided by 19. Perform the division: 19 * 217 = 4123. Subtract this product from 4131 to get the remainder: 4131 - 4123 = 8. So 4131 leaves a remainder of 8 when divided by 19. We need to add a number k such that the new remainder becomes zero. The smallest positive k that does this is k = 19 - 8 = 11. Thus 4131 + 11 = 4142, and 4142 divided by 19 gives a whole number. So the least number that must be added is 11.

Verification / Alternative Check:
We can quickly verify our result. Compute 4142 / 19. Since 19 * 200 = 3800 and 19 * 218 = 4142, the division is exact. Hence 4142 is exactly divisible by 19, confirming that adding 11 to 4131 achieves divisibility. If we test a smaller candidate such as adding 10, we get 4141. Dividing 4141 by 19 gives a remainder of 18, so 10 is not enough. This confirms that 11 is indeed the smallest integer that works.

Why Other Options Are Wrong:

• Option 10: 4131 + 10 = 4141, which is not divisible by 19 since it leaves a remainder of 18.
• Option 9: 4131 + 9 = 4140, which leaves a remainder of 17 when divided by 19.
• Option 12: 4131 + 12 = 4143, which is larger than needed and is not the least such number.
• Option 8: This is exactly the remainder of 4131 when divided by 19, but adding 8 gives 4139, which still leaves a remainder of 16. It does not make the number divisible.

Common Pitfalls:
A common error is to confuse the remainder itself with the required addition and choose 8 instead of 11. Another mistake is to perform the division incorrectly or to approximate instead of calculating the exact remainder. Some students may also test options one by one, which is slower and more error prone than using the simple formula k = d - remainder. Understanding the relationship between a number, its remainder, and the divisor is the key to solving such questions quickly.

Final Answer:
The least number that must be added to 4131 to make it divisible by 19 is 11.