Remainder transformation between related moduli A number leaves remainder 39 when divided by 357. What remainder does it leave when divided by 17?

Introduction / Context:
Converting a known remainder modulo one divisor to another is simple if the second divisor divides the first. Because congruences are compatible under such inclusion, we can reduce the known remainder directly.

Given Data / Assumptions:

• N ≡ 39 (mod 357).
• We want N mod 17.
• 357 is divisible by 17 (357 = 17 * 21).

Concept / Approach:
If N ≡ r (mod m) and k | m, then N ≡ r (mod k). Reduce r modulo k to get the canonical remainder between 0 and k − 1. No need to know the actual quotient.

Step-by-Step Solution:
Compute 39 mod 17: 17*2 = 34; remainder = 39 − 34 = 5.Therefore, N ≡ 5 (mod 17).Hence the remainder upon division by 17 is 5.

Verification / Alternative check:
Write N = 357q + 39. Modulo 17, 357q ≡ 0 (since 357 is a multiple of 17). So N ≡ 39 ≡ 5 (mod 17), confirming the result.

Why Other Options Are Wrong:
3, 6, 7 are not equal to 39 mod 17 and thus do not satisfy the congruence transformation.

Common Pitfalls:
Not recognizing that 357 is divisible by 17; mistakenly trying to compute N explicitly; arithmetic slips when reducing 39 modulo 17.

Final Answer:
5