Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 622 · 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.What is the smallest number you must add to 686 to make it divisible by 11? (a) 4 (b) 6 (c) 8 (d) 7
2. 2.What is the units digit of 3^12? (a) 9 (b) 1 (c) 7 (d) 3
3. 3.What is the units digit of 8^22? (a) 8 (b) 4 (c) 2 (d) 6
4. 4.What is the smallest number you must add to 885 to make it divisible by 11? (a) 7 (b) 6 (c) 8 (d) 5
5. 5.What is the units digit of 7^52? (a) 3 (b) 9 (c) 1 (d) 7
6. 6.What is the smallest number you must add to 301 to make it divisible by 8? (a) 5 (b) 4 (c) 3 (d) 2
7. 7.What is the remainder when 781 is divided by 12? (a) 11 (b) 0 (c) 1 (d) 2
8. 8.What is the remainder when 153 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
9. 9.What is the remainder when 566 is divided by 9? (a) 7 (b) 8 (c) 6 (d) 1
10. 10.Which of these numbers is divisible by 4? (a) 2524 (b) 2521 (c) 2526 (d) 2530
11. 11.What is the remainder when 284 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
12. 12.Which of these numbers is divisible by 3? (a) 1396 (b) 1398 (c) 1400 (d) 1402
13. 13.Which of these numbers is divisible by 11? (a) 7326 (b) 7323 (c) 7331 (d) 7346
14. 14.What is the remainder when 262 is divided by 8? (a) 2 (b) 5 (c) 6 (d) 7
15. 15.What is the remainder when 521 is divided by 9? (a) 6 (b) 8 (c) 7 (d) 1
16. 16.Which of these numbers is divisible by 6? (a) 2475 (b) 2482 (c) 2478 (d) 2487
17. 17.What is the remainder when 452 is divided by 11? (a) 1 (b) 0 (c) 10 (d) 2
18. 18.What is the units digit of 2^10? (a) 2 (b) 6 (c) 8 (d) 4
19. 19.Which of these numbers is divisible by 9? (a) 1449 (b) 1451 (c) 1465 (d) 1446
20. 20.What is the smallest number you must add to 128 to make it divisible by 3? (a) 2 (b) 1
21. 21.Which of these numbers is divisible by 8? (a) 2740 (b) 2728 (c) 2723 (d) 2729
22. 22.What is the remainder when 61 is divided by 6? (a) 2 (b) 1 (c) 0 (d) 5
23. 23.What is the remainder when 537 is divided by 8? (a) 7 (b) 1 (c) 0 (d) 2
24. 24.What is the smallest number you must add to 93 to make it divisible by 4? (a) 2 (b) 1 (c) 3
25. 25.What is the units digit of 7^40? (a) 7 (b) 9 (c) 3 (d) 1
26. 26.What is the smallest number you must add to 115 to make it divisible by 4? (a) 3 (b) 1 (c) 2
27. 27.Which of these numbers is divisible by 8? (a) 2410 (b) 2400 (c) 2401 (d) 2395
28. 28.What is the smallest number you must add to 141 to make it divisible by 4? (a) 2 (b) 1 (c) 3
29. 29.What is the remainder when 394 is divided by 8? (a) 6 (b) 1 (c) 3 (d) 2
30. 30.What is the units digit of 9^11? (a) 8 (b) 9 (c) 7 (d) 1
31. 31.Which of these numbers is divisible by 9? (a) 5139 (b) 5152 (c) 5145 (d) 5135
32. 32.Which of these numbers is divisible by 8? (a) 4160 (b) 4162 (c) 4156 (d) 4173
33. 33.What is the remainder when 331 is divided by 12? (a) 5 (b) 6 (c) 8 (d) 7
34. 34.Which of these numbers is divisible by 11? (a) 2134 (b) 2132 (c) 2151 (d) 2138
35. 35.What is the smallest number you must add to 242 to make it divisible by 9? (a) 8 (b) 1 (c) 2 (d) 3
36. 36.What is the remainder when 279 is divided by 7? (a) 5 (b) 1 (c) 4 (d) 6
37. 37.What is the units digit of 3^15? (a) 1 (b) 3 (c) 9 (d) 7
38. 38.Which of these numbers is divisible by 8? (a) 5800 (b) 5798 (c) 5813 (d) 5803
39. 39.What is the remainder when 88 is divided by 3? (a) 1 (b) 0 (c) 2
40. 40.Which of these numbers is divisible by 3? (a) 2668 (b) 2672 (c) 2667 (d) 2666

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad