Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 281 · 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 227 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 5
2. 2.What is the remainder when 184 is divided by 7? (a) 1 (b) 5 (c) 2 (d) 3
3. 3.Which of these numbers is divisible by 6? (a) 3189 (b) 3175 (c) 3180 (d) 3183
4. 4.What is the smallest number you must add to 380 to make it divisible by 9? (a) 6 (b) 8 (c) 2 (d) 7
5. 5.What is the remainder when 118 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
6. 6.Which of these numbers is divisible by 4? (a) 2275 (b) 2279 (c) 2271 (d) 2272
7. 7.What is the remainder when 97 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
8. 8.What is the smallest number you must add to 326 to make it divisible by 6? (a) 5 (b) 3 (c) 4 (d) 2
9. 9.What is the units digit of 3^59? (a) 3 (b) 9 (c) 7 (d) 1
10. 10.What is the smallest number you must add to 230 to make it divisible by 3? (a) 2 (b) 1
11. 11.What is the remainder when 363 is divided by 7? (a) 1 (b) 4 (c) 5 (d) 6
12. 12.What is the smallest number you must add to 251 to make it divisible by 12? (a) 2 (b) 3 (c) 11 (d) 1
13. 13.Which of these numbers is divisible by 11? (a) 6046 (b) 6039 (c) 6057 (d) 6034
14. 14.What is the units digit of 9^29? (a) 7 (b) 1 (c) 9 (d) 8
15. 15.What is the units digit of 2^60? (a) 2 (b) 8 (c) 6 (d) 4
16. 16.What is the remainder when 322 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
17. 17.What is the units digit of 8^46? (a) 8 (b) 4 (c) 2 (d) 6
18. 18.What is the smallest number you must add to 188 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
19. 19.Which of these numbers is divisible by 3? (a) 1400 (b) 1396 (c) 1402 (d) 1398
20. 20.What is the remainder when 143 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
21. 21.What is the units digit of 8^29? (a) 6 (b) 2 (c) 8 (d) 4
22. 22.What is the remainder when 113 is divided by 9? (a) 7 (b) 5 (c) 4 (d) 6
23. 23.What is the units digit of 4^28? (a) 7 (b) 5 (c) 4 (d) 6
24. 24.What is the smallest number you must add to 115 to make it divisible by 6? (a) 4 (b) 5 (c) 1 (d) 3
25. 25.What is the smallest number you must add to 37 to make it divisible by 3? (a) 1 (b) 2
26. 26.Which of these numbers is divisible by 8? (a) 4819 (b) 4806 (c) 4808 (d) 4815
27. 27.Which of these numbers is divisible by 3? (a) 1301 (b) 1303 (c) 1299 (d) 1297
28. 28.What is the units digit of 9^35? (a) 9 (b) 8 (c) 1 (d) 7
29. 29.What is the units digit of 8^17? (a) 8 (b) 2 (c) 4 (d) 6
30. 30.What is the units digit of 8^60? (a) 2 (b) 6 (c) 4 (d) 8
31. 31.What is the remainder when 125 is divided by 3? (a) 0 (b) 2 (c) 1
32. 32.What is the smallest number you must add to 143 to make it divisible by 10? (a) 6 (b) 7 (c) 8 (d) 3
33. 33.What is the remainder when 87 is divided by 8? (a) 7 (b) 1 (c) 6 (d) 5
34. 34.What is the smallest number you must add to 258 to make it divisible by 7? (a) 3 (b) 2 (c) 1 (d) 6
35. 35.What is the units digit of 4^60? (a) 5 (b) 6 (c) 7 (d) 4
36. 36.What is the smallest number you must add to 96 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
37. 37.What is the units digit of 9^45? (a) 8 (b) 9 (c) 1 (d) 7
38. 38.What is the smallest number you must add to 471 to make it divisible by 12? (a) 8 (b) 3 (c) 10 (d) 9
39. 39.Which of these numbers is divisible by 3? (a) 859 (b) 861 (c) 866 (d) 863
40. 40.Which of these numbers is divisible by 3? (a) 2410 (b) 2409 (c) 2408 (d) 2414

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