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

Introduction / Context:
This is a modular arithmetic question involving remainders and factors of the divisor. You are told how a number behaves when divided by 296 and then asked how it behaves when divided by 37. Since 296 and 37 are related by multiplication, we can use this relationship to transform the remainder information without knowing the original number explicitly.

Given Data / Assumptions:

• When the number N is divided by 296, the remainder is 75.
• This means N can be written as N = 296k + 75 for some integer k.
• We are asked for the remainder when N is divided by 37.
• Note that 296 = 8 × 37.

Concept / Approach:
Express the number in terms of the divisor and remainder: N = 296k + 75. Since 296 is a multiple of 37, it will leave a remainder of 0 when divided by 37. Thus, when dividing N by 37, only the remainder part 75 contributes to the final remainder. We then find the remainder when 75 is divided by 37. This is a direct application of modular arithmetic, where we use congruence to simplify the expression.

Step-by-Step Solution:
Write N in quotient remainder form with respect to 296: N = 296k + 75. Note that 296 = 8 * 37, so 296 is exactly divisible by 37. Therefore, 296k is also divisible by 37 and leaves a remainder of 0 when divided by 37. So N modulo 37 is the same as 75 modulo 37. Compute 75 divided by 37: 37 * 2 = 74, remainder 75 − 74 = 1. Thus, N leaves a remainder of 1 when divided by 37.

Verification / Alternative check:
Take a sample value for k. Let k = 1, then N = 296 * 1 + 75 = 371. Dividing 371 by 37 gives 37 * 10 = 370, remainder 1. If k = 2, then N = 296 * 2 + 75 = 667. Now 37 * 18 = 666, remainder 1 again. In each case, no matter what integer k you choose, adding 75 shifts the remainder by 75 modulo 37, always leaving 1. This confirms that the reasoning is independent of k and the result is correct.

Why Other Options Are Wrong:

• 0: This would mean N is a multiple of 37, which conflicts with N ≡ 75 (mod 296).
• 11, 8 and 19: None of these are equal to 75 modulo 37; only 1 satisfies 75 − 1 being a multiple of 37.

Common Pitfalls:
Some learners attempt to find the original number N explicitly, which is not possible with the information given and is unnecessary. Others ignore the factor relationship between 296 and 37. A good practice is to always rewrite the number in terms of quotient and remainder and then reduce modulo the new divisor, carefully using the fact that multiples of the new divisor contribute zero to the remainder.

Final Answer:
When the number is divided by 37, the remainder is 1.