A number when divided by D leaves a remainder of 8 and when divided by 3D…
2025
A number when divided by D leaves a remainder of 8 and when divided by 3D leaves a remainder of 21. What is the remainder left, when twice the number is divided by 3D?
- A.
13
- B.
3
- C.
42
- D.
Cannot be determined
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Correct answer: B
Approach: use congruences and simple algebra to find D, then compute the required remainder.
Write the number in two ways using the given remainders: n = q·D + 8 and n = r·(3D) + 21 for integers q,r.
Equate the two expressions: q·D + 8 = r·(3D) + 21. Rearranging gives D(q - 3r) = 13, so D divides 13.
Since the remainder 8 is less than D, we have D > 8, so the only possible divisor of 13 greater than 8 is D = 13. Hence 3D = 39.
Given n ≡ 21 (mod 39), compute 2n ≡ 42 ≡ 3 (mod 39). Therefore twice the number leaves remainder 3 when divided by 3D.
Answer: 3