Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 477 · 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 821 to make it divisible by 11? (a) 7 (b) 5 (c) 4 (d) 3
2. 2.What is the units digit of 2^45? (a) 4 (b) 2 (c) 8 (d) 6
3. 3.What is the units digit of 9^44? (a) 2 (b) 1 (c) 0 (d) 9
4. 4.What is the remainder when 242 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
5. 5.What is the remainder when 195 is divided by 9? (a) 6 (b) 5 (c) 7 (d) 3
6. 6.What is the remainder when 116 is divided by 5? (a) 2 (b) 0 (c) 1 (d) 4
7. 7.What is the units digit of 4^55? (a) 4 (b) 5 (c) 6 (d) 3
8. 8.Which of these numbers is divisible by 4? (a) 644 (b) 646 (c) 641 (d) 651
9. 9.What is the remainder when 224 is divided by 3? (a) 0 (b) 1 (c) 2
10. 10.What is the remainder when 335 is divided by 10? (a) 7 (b) 5 (c) 6 (d) 4
11. 11.What is the units digit of 9^41? (a) 8 (b) 9 (c) 7 (d) 1
12. 12.What is the units digit of 7^31? (a) 1 (b) 9 (c) 7 (d) 3
13. 13.What is the smallest number you must add to 316 to make it divisible by 12? (a) 9 (b) 7 (c) 4 (d) 8
14. 14.What is the remainder when 269 is divided by 8? (a) 4 (b) 3 (c) 5 (d) 6
15. 15.What is the smallest number you must add to 777 to make it divisible by 12? (a) 2 (b) 9 (c) 3 (d) 4
16. 16.Which of these numbers is divisible by 3? (a) 839 (b) 840 (c) 845 (d) 841
17. 17.What is the smallest number you must add to 383 to make it divisible by 8? (a) 3 (b) 2 (c) 1 (d) 7
18. 18.Which of these numbers is divisible by 4? (a) 872 (b) 877 (c) 875 (d) 869
19. 19.What is the smallest number you must add to 798 to make it divisible by 11? (a) 4 (b) 7 (c) 6 (d) 5
20. 20.Which of these numbers is divisible by 8? (a) 1714 (b) 1708 (c) 1726 (d) 1712
21. 21.Which of these numbers is divisible by 3? (a) 208 (b) 212 (c) 215 (d) 210
22. 22.What is the smallest number you must add to 346 to make it divisible by 11? (a) 6 (b) 7 (c) 5 (d) 8
23. 23.What is the units digit of 9^56? (a) 1 (b) 0 (c) 2 (d) 9
24. 24.What is the smallest number you must add to 260 to make it divisible by 8? (a) 5 (b) 3 (c) 4 (d) 6
25. 25.What is the remainder when 362 is divided by 11? (a) 1 (b) 10 (c) 8 (d) 9
26. 26.What is the units digit of 3^38? (a) 7 (b) 9 (c) 3 (d) 1
27. 27.What is the remainder when 749 is divided by 11? (a) 10 (b) 2 (c) 1 (d) 0
28. 28.What is the remainder when 682 is divided by 12? (a) 10 (b) 2 (c) 11 (d) 9
29. 29.Which of these numbers is divisible by 3? (a) 2406 (b) 2411 (c) 2407 (d) 2404
30. 30.What is the units digit of 8^52? (a) 2 (b) 4 (c) 8 (d) 6
31. 31.Which of these numbers is divisible by 8? (a) 5095 (b) 5082 (c) 5078 (d) 5080
32. 32.What is the units digit of 8^59? (a) 6 (b) 8 (c) 2 (d) 4
33. 33.Which of these numbers is divisible by 3? (a) 443 (b) 441 (c) 439 (d) 446
34. 34.What is the units digit of 2^25? (a) 8 (b) 6 (c) 2 (d) 4
35. 35.What is the smallest number you must add to 256 to make it divisible by 6? (a) 4 (b) 3 (c) 2 (d) 1
36. 36.What is the units digit of 7^32? (a) 1 (b) 3 (c) 7 (d) 9
37. 37.Which of these numbers is divisible by 3? (a) 363 (b) 362 (c) 367 (d) 365
38. 38.Which of these numbers is divisible by 3? (a) 1918 (b) 1921 (c) 1917 (d) 1916
39. 39.Which of these numbers is divisible by 8? (a) 5200 (b) 5203 (c) 5213 (d) 5198
40. 40.What is the smallest number you must add to 242 to make it divisible by 3? (a) 1 (b) 2

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