Discussion
A number when divided by 361 gives a remainder of 47. What remainder does it give when divided by 19?
Answer: B
Correct answer: 9
Since \( 361 = 19^{2} \), any multiple of 361 is also a multiple of 19. Write the number as \( N = 361k + 47 \). The first term is divisible by 19, so the remainder on dividing \( N \) by 19 is the remainder of 47 alone: \[ 47 = 19 \times 2 + 9, \] giving a remainder of 9. The reasoning generalises: if a number leaves remainder \( r \) on division by \( d \), and \( e \) divides \( d \), then the remainder on division by \( e \) is simply \( r \bmod e \). The condition that \( e \) divides \( d \) is essential. Without it nothing can be concluded - a number leaving remainder 47 on division by 361 could leave any remainder at all on division by, say, 7. The error to avoid is giving 47 itself, or dividing 361 rather than 47.
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