Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 532 · 40 questions

How to solve it (Patterns repeat): Divisibility: 3 and 9 — digit sum; 4 — last two digits; 8 — last three digits; 6 — even and divisible by 3; 11 — alternate digit sums differ by 0 or a multiple of 11. Remainder: find the biggest multiple not more than the number and take it away; to make a number divisible, add divisor − remainder. Units digit of a power: write the cycle (7, 9, 3, 1 for 7), divide the power by the cycle length and use the remainder.

1. 1.Which of these numbers is divisible by 4? (a) 1761 (b) 1757 (c) 1760 (d) 1765
2. 2.What is the remainder when 62 is divided by 3? (a) 0 (b) 1 (c) 2
3. 3.What is the smallest number you must add to 323 to make it divisible by 11? (a) 6 (b) 4 (c) 8 (d) 7
4. 4.Which of these numbers is divisible by 9? (a) 570 (b) 558 (c) 564 (d) 555
5. 5.What is the smallest number you must add to 679 to make it divisible by 10? (a) 3 (b) 9 (c) 2 (d) 1
6. 6.What is the remainder when 170 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 2
7. 7.Which of these numbers is divisible by 6? (a) 503 (b) 493 (c) 490 (d) 492
8. 8.Which of these numbers is divisible by 8? (a) 5302 (b) 5290 (c) 5288 (d) 5286
9. 9.What is the units digit of 4^33? (a) 3 (b) 6 (c) 5 (d) 4
10. 10.What is the units digit of 2^11? (a) 8 (b) 6 (c) 2 (d) 4
11. 11.What is the units digit of 4^23? (a) 6 (b) 3 (c) 4 (d) 5
12. 12.Which of these numbers is divisible by 9? (a) 6312 (b) 6309 (c) 6320 (d) 6305
13. 13.What is the smallest number you must add to 88 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
14. 14.What is the remainder when 619 is divided by 11? (a) 8 (b) 2 (c) 4 (d) 3
15. 15.What is the units digit of 3^46? (a) 9 (b) 1 (c) 7 (d) 3
16. 16.What is the smallest number you must add to 135 to make it divisible by 12? (a) 8 (b) 10 (c) 9 (d) 3
17. 17.Which of these numbers is divisible by 9? (a) 1125 (b) 1126 (c) 1121 (d) 1137
18. 18.Which of these numbers is divisible by 3? (a) 1219 (b) 1213 (c) 1216 (d) 1215
19. 19.Which of these numbers is divisible by 3? (a) 1622 (b) 1623 (c) 1624 (d) 1627
20. 20.What is the smallest number you must add to 309 to make it divisible by 10? (a) 2 (b) 1 (c) 9 (d) 3
21. 21.What is the remainder when 87 is divided by 7? (a) 5 (b) 2 (c) 3 (d) 4
22. 22.Which of these numbers is divisible by 3? (a) 911 (b) 906 (c) 904 (d) 907
23. 23.What is the units digit of 7^12? (a) 7 (b) 1 (c) 3 (d) 9
24. 24.What is the remainder when 424 is divided by 9? (a) 2 (b) 8 (c) 0 (d) 1
25. 25.Which of these numbers is divisible by 4? (a) 565 (b) 562 (c) 569 (d) 564
26. 26.Which of these numbers is divisible by 8? (a) 1119 (b) 1125 (c) 1130 (d) 1120
27. 27.Which of these numbers is divisible by 4? (a) 1158 (b) 1161 (c) 1166 (d) 1160
28. 28.Which of these numbers is divisible by 3? (a) 1771 (b) 1766 (c) 1767 (d) 1768
29. 29.What is the remainder when 484 is divided by 9? (a) 8 (b) 7 (c) 6 (d) 2
30. 30.Which of these numbers is divisible by 8? (a) 6216 (b) 6220 (c) 6215 (d) 6227
31. 31.What is the units digit of 3^60? (a) 9 (b) 3 (c) 7 (d) 1
32. 32.What is the units digit of 9^17? (a) 7 (b) 9 (c) 1 (d) 8
33. 33.Which of these numbers is divisible by 4? (a) 1191 (b) 1194 (c) 1192 (d) 1197
34. 34.What is the remainder when 351 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 0
35. 35.Which of these numbers is divisible by 3? (a) 2528 (b) 2533 (c) 2530 (d) 2529
36. 36.What is the smallest number you must add to 724 to make it divisible by 12? (a) 4 (b) 9 (c) 8 (d) 7
37. 37.What is the units digit of 9^10? (a) 2 (b) 9 (c) 1 (d) 0
38. 38.Which of these numbers is divisible by 6? (a) 1072 (b) 1079 (c) 1082 (d) 1074
39. 39.Which of these numbers is divisible by 6? (a) 3023 (b) 3017 (c) 3029 (d) 3018
40. 40.Which of these numbers is divisible by 9? (a) 5508 (b) 5507 (c) 5510 (d) 5519

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