Multiple choice general knowledge math & puzzles

Z is a set of three digit numbers of base 10 which are divisible by 3 and end with 1 when written in base 5 and base 7 notations, then how many elements does Z has?

  1. 8

  2. 9

  3. 10

  4. 12

Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Find three-digit numbers where: divisible by 3, ends in 1 in base 5 (so ≡ 1 mod 5), ends in 1 in base 7 (so ≡ 1 mod 7). By CRT, x ≡ 1 (mod 35). Three-digit candidates: 106, 141, 176, 211, 246, 281, 316, 351, 386, 421, 456, 491, 526, 561, 596, 631, 666, 701, 736, 771, 806, 841, 876, 911, 946, 981. Only those divisible by 3 are valid: 141, 81, 246, 351, 561, 666, 841, 946 - totaling 9 numbers.