A number when divided by 779 leaves a remainder of 47. If the same number is divided by 19, what remainder will it leave?

Introduction:
This question tests your understanding of modular arithmetic and remainders. You are given the remainder when a number is divided by a large divisor and asked to find the remainder when the same number is divided by a smaller related divisor.

Given Data / Assumptions:

• Let the unknown number be N.
• When N is divided by 779, the remainder is 47.
• We need the remainder when N is divided by 19.
• All divisions are in the standard integer sense.

Concept / Approach:
We express N in terms of the first division and then reduce modulo 19. Also, we notice that 779 is a multiple of 19, which greatly simplifies the calculation because any multiple of 779 is automatically a multiple of 19.

Step-by-Step Solution:
Step 1: Express N using the given remainder.N = 779k + 47 for some integer k.Step 2: Note that 779 = 19 * 41.Therefore, 779k = 19 * 41k is divisible by 19.Step 3: Reduce N modulo 19.N mod 19 = (779k + 47) mod 19.Since 779k is divisible by 19, it contributes remainder 0.So N mod 19 = 47 mod 19.Step 4: Compute 47 mod 19.19 * 2 = 38, 47 − 38 = 9.So the remainder is 9.

Verification / Alternative check:
You can pick a random k, say k = 1: N = 779 + 47 = 826. Now 826 ÷ 19 = 43 remainder 9. For any other integer k, the term 779k always adds a multiple of 19, so the remainder will stay 9.

Why Other Options Are Wrong:
Remainders 6, 5, 8 and 3 are not equal to 47 mod 19. Only 9 matches the correct reduction of 47 modulo 19.

Common Pitfalls:
Some students try to work directly with the large number 779, forgetting to factor it or see its relation to 19. Others mistakenly divide 779 by 19 and then mishandle the remainder instead of using the expression N = 779k + 47.

Final Answer:
When the number is divided by 19, the remainder is 9.