When a positive integer x is divided by a certain divisor, the divisor is equal to 4 times the quotient and also equal to double the remainder. If the remainder in this division is 80, what is the value of x (the original number)?

Introduction / Context:
This quantitative aptitude problem checks understanding of the division algorithm, which connects dividend, divisor, quotient, and remainder. Instead of directly giving the divisor and quotient, the question describes relationships between these three quantities in words. The task is to translate those relationships into equations and then find the original number x. This type of question is very common in banking, SSC, and other competitive exams.

Given Data / Assumptions:

• x is a positive integer.
• Divisor = 4 times the quotient.
• Divisor = 2 times the remainder.
• The remainder is 80.
• Standard division algorithm: dividend = divisor * quotient + remainder.

Concept / Approach:
The key idea is to express everything in terms of the divisor, quotient, and remainder. Once we know two independent equations relating these three quantities, we can find the divisor and quotient. After that, we apply the division algorithm to compute x. The division algorithm says that for any integer x divided by a divisor d, with quotient q and remainder r, we have x = d * q + r, where 0 ≤ r < d.

Step-by-Step Solution:
Given remainder r = 80.Divisor d is double the remainder, so d = 2 * r = 2 * 80 = 160.The divisor is also 4 times the quotient, so d = 4 * q.Substitute d = 160 into d = 4 * q to get 160 = 4 * q.Therefore, q = 160 / 4 = 40.Now use the division algorithm: x = d * q + r.So x = 160 * 40 + 80.Compute 160 * 40 = 6400.Thus x = 6400 + 80 = 6480.

Verification / Alternative check:
Check remainder: 6480 / 160 gives quotient 40 and remainder 80 because 160 * 40 = 6400 and 6480 - 6400 = 80.Check relations: divisor d = 160, quotient q = 40, remainder r = 80.Verify d = 4 * q: 4 * 40 = 160, which matches d.Verify d = 2 * r: 2 * 80 = 160, which matches d.All conditions are satisfied, so x = 6480 is consistent.

Why Other Options Are Wrong:
9680 gives a different remainder or quotient when divided by 160, so it does not satisfy both divisor relations.8460 does not produce divisor equal to 4 times the quotient and 2 times the remainder simultaneously.4680 also fails the required conditions when tested with the possible divisor 160.

Common Pitfalls:
One common mistake is to assume divisor equals remainder or quotient instead of using the precise relations divisor = 4 * quotient and divisor = 2 * remainder.Another frequent error is writing the division formula incorrectly, for example x = divisor + quotient + remainder instead of x = divisor * quotient + remainder.Students may also forget that the same divisor value must satisfy both given relations at the same time.

Final Answer:
The value of the number x that satisfies all the given conditions is 6480.