Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 468 · 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 327 to make it divisible by 9? (a) 6 (b) 5 (c) 7 (d) 3
2. 2.What is the units digit of 2^38? (a) 4 (b) 2 (c) 8 (d) 6
3. 3.Which of these numbers is divisible by 6? (a) 1503 (b) 1494 (c) 1491 (d) 1498
4. 4.Which of these numbers is divisible by 3? (a) 611 (b) 607 (c) 609 (d) 614
5. 5.What is the units digit of 4^26? (a) 7 (b) 6 (c) 5 (d) 4
6. 6.Which of these numbers is divisible by 8? (a) 2658 (b) 2664 (c) 2667 (d) 2674
7. 7.Which of these numbers is divisible by 3? (a) 2615 (b) 2613 (c) 2611 (d) 2618
8. 8.What is the units digit of 8^22? (a) 2 (b) 4 (c) 6 (d) 8
9. 9.What is the units digit of 7^19? (a) 3 (b) 7 (c) 9 (d) 1
10. 10.What is the smallest number you must add to 99 to make it divisible by 4? (a) 1 (b) 2 (c) 3
11. 11.What is the smallest number you must add to 388 to make it divisible by 11? (a) 8 (b) 7 (c) 3 (d) 9
12. 12.What is the units digit of 7^11? (a) 7 (b) 1 (c) 3 (d) 9
13. 13.What is the remainder when 150 is divided by 8? (a) 2 (b) 5 (c) 7 (d) 6
14. 14.Which of these numbers is divisible by 9? (a) 3321 (b) 3337 (c) 3322 (d) 3319
15. 15.What is the remainder when 154 is divided by 9? (a) 8 (b) 0 (c) 2 (d) 1
16. 16.Which of these numbers is divisible by 11? (a) 2212 (b) 2230 (c) 2202 (d) 2211
17. 17.What is the units digit of 3^20? (a) 7 (b) 1 (c) 9 (d) 3
18. 18.What is the remainder when 301 is divided by 11? (a) 7 (b) 4 (c) 3 (d) 5
19. 19.Which of these numbers is divisible by 8? (a) 2912 (b) 2905 (c) 2922 (d) 2915
20. 20.What is the units digit of 8^30? (a) 8 (b) 4 (c) 2 (d) 6
21. 21.What is the smallest number you must add to 65 to make it divisible by 3? (a) 1 (b) 2
22. 22.What is the remainder when 397 is divided by 12? (a) 0 (b) 2 (c) 11 (d) 1
23. 23.What is the remainder when 517 is divided by 8? (a) 4 (b) 6 (c) 5 (d) 3
24. 24.What is the units digit of 3^50? (a) 1 (b) 7 (c) 9 (d) 3
25. 25.Which of these numbers is divisible by 3? (a) 240 (b) 239 (c) 245 (d) 241
26. 26.Which of these numbers is divisible by 4? (a) 1721 (b) 1716 (c) 1719 (d) 1713
27. 27.What is the remainder when 680 is divided by 9? (a) 6 (b) 7 (c) 4 (d) 5
28. 28.What is the smallest number you must add to 230 to make it divisible by 3? (a) 1 (b) 2
29. 29.What is the remainder when 139 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
30. 30.What is the smallest number you must add to 338 to make it divisible by 7? (a) 4 (b) 5 (c) 6 (d) 2
31. 31.Which of these numbers is divisible by 6? (a) 853 (b) 847 (c) 846 (d) 845
32. 32.What is the units digit of 9^15? (a) 8 (b) 9 (c) 1 (d) 7
33. 33.What is the smallest number you must add to 217 to make it divisible by 8? (a) 6 (b) 1 (c) 5 (d) 7
34. 34.Which of these numbers is divisible by 9? (a) 3896 (b) 3886 (c) 3898 (d) 3888
35. 35.What is the remainder when 257 is divided by 12? (a) 6 (b) 7 (c) 4 (d) 5
36. 36.Which of these numbers is divisible by 11? (a) 1026 (b) 1038 (c) 1023 (d) 1020
37. 37.Which of these numbers is divisible by 4? (a) 2027 (b) 2019 (c) 2023 (d) 2020
38. 38.Which of these numbers is divisible by 6? (a) 5360 (b) 5347 (c) 5357 (d) 5352
39. 39.Which of these numbers is divisible by 9? (a) 2743 (b) 2731 (c) 2747 (d) 2736
40. 40.Which of these numbers is divisible by 9? (a) 2369 (b) 2350 (c) 2358 (d) 2361

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