Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 870 · 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 9^35? (a) 9 (b) 8 (c) 7 (d) 1
2. 2.What is the smallest number you must add to 586 to make it divisible by 11? (a) 8 (b) 9 (c) 7 (d) 3
3. 3.What is the units digit of 2^56? (a) 6 (b) 8 (c) 2 (d) 4
4. 4.What is the smallest number you must add to 182 to make it divisible by 6? (a) 5 (b) 4 (c) 2 (d) 3
5. 5.What is the units digit of 4^23? (a) 3 (b) 5 (c) 4 (d) 6
6. 6.Which of these numbers is divisible by 4? (a) 2198 (b) 2200 (c) 2205 (d) 2202
7. 7.What is the remainder when 241 is divided by 7? (a) 4 (b) 5 (c) 3 (d) 2
8. 8.What is the smallest number you must add to 118 to make it divisible by 7? (a) 1 (b) 3 (c) 2 (d) 6
9. 9.Which of these numbers is divisible by 6? (a) 3627 (b) 3618 (c) 3622 (d) 3617
10. 10.Which of these numbers is divisible by 6? (a) 2676 (b) 2677 (c) 2686 (d) 2675
11. 11.Which of these numbers is divisible by 11? (a) 8043 (b) 8032 (c) 8041 (d) 8055
12. 12.Which of these numbers is divisible by 3? (a) 584 (b) 579 (c) 581 (d) 578
13. 13.What is the units digit of 8^55? (a) 6 (b) 8 (c) 2 (d) 4
14. 14.What is the units digit of 4^47? (a) 5 (b) 4 (c) 3 (d) 6
15. 15.Which of these numbers is divisible by 6? (a) 1027 (b) 1036 (c) 1042 (d) 1032
16. 16.Which of these numbers is divisible by 11? (a) 7307 (b) 7301 (c) 7293 (d) 7284
17. 17.What is the remainder when 231 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 5
18. 18.What is the units digit of 8^50? (a) 8 (b) 2 (c) 4 (d) 6
19. 19.What is the remainder when 753 is divided by 11? (a) 4 (b) 6 (c) 7 (d) 5
20. 20.Which of these numbers is divisible by 9? (a) 7165 (b) 7155 (c) 7157 (d) 7147
21. 21.What is the remainder when 350 is divided by 8? (a) 5 (b) 7 (c) 2 (d) 6
22. 22.What is the smallest number you must add to 165 to make it divisible by 4? (a) 1 (b) 3 (c) 2
23. 23.What is the smallest number you must add to 387 to make it divisible by 8? (a) 5 (b) 3 (c) 4 (d) 6
24. 24.What is the smallest number you must add to 83 to make it divisible by 7? (a) 1 (b) 3 (c) 2 (d) 6
25. 25.What is the units digit of 3^42? (a) 1 (b) 3 (c) 7 (d) 9
26. 26.Which of these numbers is divisible by 6? (a) 1206 (b) 1217 (c) 1204 (d) 1207
27. 27.What is the units digit of 8^49? (a) 2 (b) 8 (c) 4 (d) 6
28. 28.Which of these numbers is divisible by 11? (a) 8337 (b) 8348 (c) 8357 (d) 8338
29. 29.What is the units digit of 4^41? (a) 4 (b) 3 (c) 6 (d) 5
30. 30.What is the units digit of 3^58? (a) 7 (b) 9 (c) 1 (d) 3
31. 31.Which of these numbers is divisible by 11? (a) 2310 (b) 2300 (c) 2327 (d) 2315
32. 32.What is the remainder when 200 is divided by 3? (a) 1 (b) 0 (c) 2
33. 33.What is the units digit of 8^25? (a) 6 (b) 4 (c) 2 (d) 8
34. 34.Which of these numbers is divisible by 3? (a) 1212 (b) 1217 (c) 1211 (d) 1213
35. 35.Which of these numbers is divisible by 11? (a) 8492 (b) 8482 (c) 8505 (d) 8496
36. 36.Which of these numbers is divisible by 4? (a) 1370 (b) 1377 (c) 1375 (d) 1372
37. 37.What is the remainder when 754 is divided by 12? (a) 2 (b) 11 (c) 9 (d) 10
38. 38.Which of these numbers is divisible by 11? (a) 1274 (b) 1244 (c) 1254 (d) 1264
39. 39.What is the smallest number you must add to 332 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
40. 40.What is the remainder when 694 is divided by 10? (a) 6 (b) 4 (c) 3 (d) 5

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