A number leaves a remainder of 75 when divided by 296. When the same number is divided by 37, what remainder will be obtained?

Introduction / Context:
This question is about remainders and the relationship between two divisors that share a common factor. We are told how a number behaves when divided by a larger divisor and then asked how it behaves when divided by a related smaller divisor. Such problems are standard in number system and modular arithmetic questions in aptitude exams.

Given Data / Assumptions:
Let the unknown number be N.When N is divided by 296, the remainder is 75.We need the remainder when N is divided by 37.

Concept / Approach:
Since 296 and 37 are related by a simple multiplication factor, we can express the divisibility relation using the equation N = 296k + 75 for some integer k. Noticing that 296 = 37 * 8 allows us to rewrite N in terms of 37. Once N is expressed in the form 37m + r, the remainder r when dividing by 37 becomes clear. This uses basic modular arithmetic and factorization.

Step-by-Step Solution:
Step 1: Express the given remainder condition: N = 296k + 75 for some integer k.Step 2: Factorize 296 as 296 = 37 * 8.Step 3: Substitute into the expression for N: N = 37 * 8k + 75.Step 4: We want to express N in the form 37m + r, where r is the remainder when N is divided by 37.Step 5: Note that 75 can be written as 37 * 2 + 1 because 37 * 2 = 74, leaving remainder 1.Step 6: So N = 37 * 8k + 37 * 2 + 1 = 37 * (8k + 2) + 1.Step 7: Therefore, when N is divided by 37, the quotient is (8k + 2) and the remainder is 1.

Verification / Alternative check:
We can use a concrete value to test the logic. For example, if k = 1, then N would be 296 * 1 + 75 = 371. Dividing 371 by 296 indeed leaves remainder 75. Now divide 371 by 37: 37 * 10 = 370 and the remainder is 1. This agrees with our general result. Changing k will not affect the remainder when dividing by 37 because 296k is always a multiple of 37.

Why Other Options Are Wrong:
The remainders 2, 3 and 5 would require expressing 75 as 37 times some integer plus those remainders, but 75 − 2, 75 − 3 and 75 − 5 are not multiples of 37. Only 75 − 1 = 74 is exactly divisible by 37. Therefore, these distractor options contradict the factorization argument and numerical checks.

Common Pitfalls:
One typical mistake is to attempt long division directly with large numbers instead of exploiting the relationship between divisors. Another is to forget that 296 is a multiple of 37, which makes the simplification much easier. Recognizing factorization patterns is crucial for solving such questions efficiently.

Final Answer:
The remainder when the number is divided by 37 is 1.