The difference between two positive integers is 1146. When the larger integer is divided by the smaller integer, the quotient is 4 and the remainder is 6. What is the value of the larger integer?

Introduction / Context:
This question tests understanding of division with remainder and the relationship between two integers when both the difference and the quotient with remainder are given. Problems of this type are common in number system sections of aptitude exams and require converting the language into algebraic equations.

Given Data / Assumptions:

• There are two positive integers, a larger one L and a smaller one S.
• The difference between them is L − S = 1146.
• When L is divided by S, the quotient is 4 and the remainder is 6.
• By the division algorithm, L = 4S + 6.

Concept / Approach:
We use the division algorithm and the given difference to form a system of two equations in L and S. The division algorithm states that dividend = divisor × quotient + remainder. Together with the difference equation, we can eliminate one variable and solve for the other. Once we know the smaller integer S, we substitute back to find L.

Step-by-Step Solution:
From the difference condition, we have L − S = 1146.From the division condition, L = 4S + 6.Substitute L = 4S + 6 into L − S = 1146.This gives (4S + 6) − S = 1146.Simplify: 3S + 6 = 1146.Subtract 6 from both sides: 3S = 1140.Divide by 3: S = 1140 / 3 = 380.Now compute L using L = 4S + 6: L = 4 × 380 + 6 = 1520 + 6 = 1526.

Verification / Alternative check:
Check the difference: L − S = 1526 − 380 = 1146, which matches the given information. Now divide 1526 by 380. We get 380 × 4 = 1520 with remainder 6, so the quotient is 4 and the remainder is 6. Both conditions are satisfied, confirming that 1526 is correct for the larger integer.

Why Other Options Are Wrong:
1431 minus its corresponding smaller integer would not yield 1146 while still giving quotient 4 and remainder 6. 1485 and 1606 also fail at least one of the two checks: either the difference is not 1146 or the quotient and remainder are not 4 and 6 respectively. 1234 is too small to accommodate the required difference with a positive smaller integer. Only 1526 satisfies both conditions simultaneously.

Common Pitfalls:
Students sometimes reverse the roles of the larger and smaller integers, or they forget to use the remainder correctly in the division algorithm. Another frequent error is to misread L − S as S − L, which changes the sign and leads to a negative smaller integer. Writing down both equations clearly and substituting carefully avoids these misunderstandings.

Final Answer:
The value of the larger integer is 1526.