Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 466 · 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 remainder when 212 is divided by 11? (a) 8 (b) 3 (c) 4 (d) 2
2. 2.What is the remainder when 503 is divided by 7? (a) 1 (b) 6 (c) 5 (d) 4
3. 3.What is the units digit of 4^40? (a) 4 (b) 7 (c) 5 (d) 6
4. 4.What is the remainder when 218 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
5. 5.Which of these numbers is divisible by 3? (a) 2377 (b) 2374 (c) 2376 (d) 2380
6. 6.What is the remainder when 669 is divided by 11? (a) 8 (b) 9 (c) 10 (d) 2
7. 7.What is the remainder when 803 is divided by 12? (a) 10 (b) 1 (c) 9 (d) 11
8. 8.What is the smallest number you must add to 567 to make it divisible by 10? (a) 2 (b) 4 (c) 3 (d) 7
9. 9.What is the units digit of 9^58? (a) 1 (b) 9 (c) 0 (d) 2
10. 10.What is the units digit of 7^16? (a) 3 (b) 1 (c) 9 (d) 7
11. 11.What is the units digit of 2^59? (a) 2 (b) 8 (c) 6 (d) 4
12. 12.What is the smallest number you must add to 259 to make it divisible by 11? (a) 7 (b) 5 (c) 6 (d) 4
13. 13.What is the smallest number you must add to 365 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 1
14. 14.What is the units digit of 2^21? (a) 8 (b) 4 (c) 6 (d) 2
15. 15.Which of these numbers is divisible by 9? (a) 1017 (b) 1033 (c) 1020 (d) 1014
16. 16.What is the smallest number you must add to 193 to make it divisible by 3? (a) 1 (b) 2
17. 17.Which of these numbers is divisible by 8? (a) 6812 (b) 6800 (c) 6807 (d) 6798
18. 18.What is the smallest number you must add to 249 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
19. 19.What is the units digit of 4^32? (a) 6 (b) 4 (c) 7 (d) 5
20. 20.What is the units digit of 4^26? (a) 7 (b) 4 (c) 5 (d) 6
21. 21.What is the smallest number you must add to 333 to make it divisible by 7? (a) 4 (b) 2 (c) 5 (d) 3
22. 22.What is the units digit of 7^31? (a) 3 (b) 9 (c) 1 (d) 7
23. 23.What is the remainder when 220 is divided by 3? (a) 0 (b) 1 (c) 2
24. 24.What is the units digit of 2^13? (a) 6 (b) 8 (c) 4 (d) 2
25. 25.What is the remainder when 237 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
26. 26.What is the smallest number you must add to 125 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 1
27. 27.What is the units digit of 8^45? (a) 8 (b) 2 (c) 6 (d) 4
28. 28.What is the remainder when 818 is divided by 12? (a) 10 (b) 1 (c) 3 (d) 2
29. 29.Which of these numbers is divisible by 9? (a) 3124 (b) 3123 (c) 3140 (d) 3120
30. 30.Which of these numbers is divisible by 4? (a) 854 (b) 863 (c) 856 (d) 859
31. 31.What is the smallest number you must add to 241 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
32. 32.What is the units digit of 9^40? (a) 1 (b) 9 (c) 0 (d) 2
33. 33.Which of these numbers is divisible by 4? (a) 2981 (b) 2975 (c) 2976 (d) 2979
34. 34.What is the remainder when 313 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
35. 35.What is the smallest number you must add to 413 to make it divisible by 10? (a) 8 (b) 3 (c) 6 (d) 7
36. 36.What is the units digit of 7^42? (a) 1 (b) 7 (c) 9 (d) 3
37. 37.What is the smallest number you must add to 234 to make it divisible by 4? (a) 2 (b) 1 (c) 3
38. 38.Which of these numbers is divisible by 11? (a) 4312 (b) 4329 (c) 4314 (d) 4305
39. 39.Which of these numbers is divisible by 6? (a) 1096 (b) 1092 (c) 1087 (d) 1102
40. 40.What is the units digit of 3^21? (a) 7 (b) 3 (c) 1 (d) 9

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