A number leaves a remainder of 75 when divided by 296. What remainder will the same number leave when it is divided by 37?

Introduction:
This is a modular arithmetic question involving remainders and factors. You are given the remainder when a number is divided by a larger divisor and are asked to find the remainder with respect to a smaller divisor that is a factor of the larger one.

Given Data / Assumptions:

• Let the number be N.
• N leaves a remainder of 75 when divided by 296.
• We need the remainder when N is divided by 37.
• Note that 296 is a multiple of 37.

Concept / Approach:
If a number N can be written as: N = 296k + 75 for some integer k, and if 296 is a multiple of 37, then: N mod 37 = (296k + 75) mod 37 = (296k mod 37) + (75 mod 37). Since 296k is exactly divisible by 37, the remainder will be determined entirely by 75 mod 37.

Step-by-Step Solution:
Step 1: Express N using the given remainder. N = 296k + 75. Step 2: Express 296 in terms of 37. 296 = 37 * 8. So: N = 37 * 8k + 75. Step 3: Compute N modulo 37. Because 37 * 8k is divisible by 37, its remainder is 0: N mod 37 = 75 mod 37. Compute 75 mod 37: 37 * 2 = 74. 75 - 74 = 1. So: 75 mod 37 = 1. Therefore: N mod 37 = 1.

Verification / Alternative Check:
Take one specific value of k, say k = 1: N = 296 * 1 + 75 = 371. Now divide 371 by 37: 37 * 10 = 370, remainder = 1. This confirms that the remainder is indeed 1.

Why Other Options Are Wrong:
0, 8, 11, and 19 do not match 75 mod 37, and any correct approach using 296 = 37 * 8 will always lead to 1 as the remainder. Therefore these other options are incorrect.

Common Pitfalls:
A common error is to attempt long division directly with unknown N instead of using modular properties. Another mistake is to assume 75 is itself the remainder when divided by 37 without performing the final reduction to a value less than 37.

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