Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 335 · 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 units digit of 3^22? (a) 3 (b) 1 (c) 9 (d) 7
2. 2.Which of these numbers is divisible by 9? (a) 4310 (b) 4293 (c) 4298 (d) 4290
3. 3.What is the smallest number you must add to 130 to make it divisible by 9? (a) 7 (b) 5 (c) 4 (d) 6
4. 4.What is the remainder when 121 is divided by 3? (a) 1 (b) 0 (c) 2
5. 5.Which of these numbers is divisible by 3? (a) 308 (b) 313 (c) 310 (d) 309
6. 6.Which of these numbers is divisible by 11? (a) 9264 (b) 9251 (c) 9257 (d) 9247
7. 7.Which of these numbers is divisible by 9? (a) 4931 (b) 4943 (c) 4938 (d) 4932
8. 8.What is the smallest number you must add to 71 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
9. 9.What is the smallest number you must add to 597 to make it divisible by 10? (a) 2 (b) 7 (c) 4 (d) 3
10. 10.Which of these numbers is divisible by 4? (a) 1846 (b) 1850 (c) 1844 (d) 1843
11. 11.Which of these numbers is divisible by 9? (a) 3393 (b) 3387 (c) 3394 (d) 3407
12. 12.What is the smallest number you must add to 601 to make it divisible by 8? (a) 7 (b) 1 (c) 6 (d) 5
13. 13.What is the smallest number you must add to 397 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 5
14. 14.Which of these numbers is divisible by 3? (a) 1433 (b) 1436 (c) 1438 (d) 1434
15. 15.What is the remainder when 611 is divided by 8? (a) 5 (b) 2 (c) 4 (d) 3
16. 16.Which of these numbers is divisible by 11? (a) 2857 (b) 2866 (c) 2860 (d) 2876
17. 17.What is the units digit of 3^51? (a) 7 (b) 1 (c) 9 (d) 3
18. 18.Which of these numbers is divisible by 3? (a) 711 (b) 715 (c) 712 (d) 710
19. 19.Which of these numbers is divisible by 3? (a) 832 (b) 835 (c) 831 (d) 829
20. 20.What is the remainder when 235 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
21. 21.What is the smallest number you must add to 161 to make it divisible by 4? (a) 2 (b) 3 (c) 1
22. 22.Which of these numbers is divisible by 9? (a) 3126 (b) 3142 (c) 3139 (d) 3132
23. 23.What is the remainder when 164 is divided by 7? (a) 5 (b) 2 (c) 3 (d) 4
24. 24.What is the remainder when 634 is divided by 10? (a) 6 (b) 5 (c) 3 (d) 4
25. 25.What is the smallest number you must add to 322 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
26. 26.What is the units digit of 4^42? (a) 7 (b) 5 (c) 4 (d) 6
27. 27.What is the smallest number you must add to 65 to make it divisible by 6? (a) 5 (b) 1 (c) 3 (d) 2
28. 28.What is the remainder when 604 is divided by 8? (a) 6 (b) 4 (c) 5 (d) 3
29. 29.What is the units digit of 3^56? (a) 9 (b) 1 (c) 3 (d) 7
30. 30.What is the units digit of 8^58? (a) 2 (b) 8 (c) 6 (d) 4
31. 31.What is the remainder when 56 is divided by 3? (a) 2 (b) 1 (c) 0
32. 32.What is the units digit of 3^10? (a) 7 (b) 1 (c) 3 (d) 9
33. 33.What is the units digit of 3^41? (a) 1 (b) 7 (c) 3 (d) 9
34. 34.What is the smallest number you must add to 176 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
35. 35.What is the units digit of 9^16? (a) 1 (b) 0 (c) 9 (d) 2
36. 36.What is the units digit of 2^37? (a) 8 (b) 2 (c) 6 (d) 4
37. 37.What is the units digit of 2^51? (a) 6 (b) 4 (c) 2 (d) 8
38. 38.What is the smallest number you must add to 496 to make it divisible by 11? (a) 1 (b) 9 (c) 8 (d) 10
39. 39.What is the remainder when 496 is divided by 7? (a) 6 (b) 1 (c) 4 (d) 5
40. 40.Which of these numbers is divisible by 4? (a) 2505 (b) 2504 (c) 2509 (d) 2501

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