Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 911 · 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 2^22? (a) 2 (b) 8 (c) 4 (d) 6
2. 2.Which of these numbers is divisible by 4? (a) 1366 (b) 1373 (c) 1368 (d) 1371
3. 3.What is the units digit of 8^28? (a) 8 (b) 2 (c) 4 (d) 6
4. 4.What is the units digit of 9^22? (a) 0 (b) 9 (c) 1 (d) 2
5. 5.What is the units digit of 9^60? (a) 9 (b) 0 (c) 2 (d) 1
6. 6.What is the units digit of 2^10? (a) 8 (b) 2 (c) 6 (d) 4
7. 7.What is the remainder when 266 is divided by 12? (a) 2 (b) 1 (c) 10 (d) 3
8. 8.What is the smallest number you must add to 303 to make it divisible by 9? (a) 6 (b) 2 (c) 4 (d) 3
9. 9.Which of these numbers is divisible by 4? (a) 2998 (b) 3002 (c) 3000 (d) 3005
10. 10.Which of these numbers is divisible by 9? (a) 2628 (b) 2642 (c) 2630 (d) 2621
11. 11.Which of these numbers is divisible by 4? (a) 2293 (b) 2288 (c) 2291 (d) 2286
12. 12.What is the remainder when 274 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
13. 13.What is the smallest number you must add to 164 to make it divisible by 7? (a) 3 (b) 4 (c) 6 (d) 5
14. 14.What is the remainder when 295 is divided by 6? (a) 1 (b) 0 (c) 2 (d) 5
15. 15.Which of these numbers is divisible by 6? (a) 4136 (b) 4140 (c) 4145 (d) 4151
16. 16.What is the remainder when 246 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
17. 17.What is the smallest number you must add to 202 to make it divisible by 3? (a) 1 (b) 2
18. 18.What is the remainder when 195 is divided by 6? (a) 2 (b) 5 (c) 4 (d) 3
19. 19.What is the remainder when 193 is divided by 3? (a) 0 (b) 1 (c) 2
20. 20.What is the units digit of 8^34? (a) 4 (b) 8 (c) 6 (d) 2
21. 21.What is the smallest number you must add to 634 to make it divisible by 9? (a) 5 (b) 7 (c) 4 (d) 6
22. 22.Which of these numbers is divisible by 11? (a) 575 (b) 590 (c) 571 (d) 572
23. 23.What is the remainder when 402 is divided by 8? (a) 3 (b) 1 (c) 2 (d) 6
24. 24.What is the remainder when 597 is divided by 11? (a) 8 (b) 4 (c) 3 (d) 2
25. 25.What is the smallest number you must add to 66 to make it divisible by 4? (a) 2 (b) 1 (c) 3
26. 26.What is the remainder when 157 is divided by 10? (a) 8 (b) 3 (c) 7 (d) 6
27. 27.Which of these numbers is divisible by 11? (a) 7510 (b) 7528 (c) 7513 (d) 7514
28. 28.Which of these numbers is divisible by 9? (a) 3281 (b) 3291 (c) 3276 (d) 3270
29. 29.What is the remainder when 179 is divided by 6? (a) 1 (b) 5 (c) 4 (d) 3
30. 30.What is the remainder when 326 is divided by 12? (a) 2 (b) 3 (c) 10 (d) 1
31. 31.What is the smallest number you must add to 376 to make it divisible by 10? (a) 3 (b) 4 (c) 5 (d) 6
32. 32.What is the smallest number you must add to 330 to make it divisible by 7? (a) 6 (b) 1 (c) 4 (d) 5
33. 33.What is the smallest number you must add to 61 to make it divisible by 6? (a) 3 (b) 1 (c) 4 (d) 5
34. 34.What is the units digit of 2^18? (a) 8 (b) 2 (c) 6 (d) 4
35. 35.What is the remainder when 299 is divided by 10? (a) 8 (b) 9 (c) 1 (d) 7
36. 36.What is the units digit of 7^24? (a) 7 (b) 9 (c) 1 (d) 3
37. 37.What is the remainder when 167 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
38. 38.What is the smallest number you must add to 966 to make it divisible by 12? (a) 6 (b) 7 (c) 8 (d) 5
39. 39.What is the remainder when 131 is divided by 5? (a) 0 (b) 4 (c) 1 (d) 2
40. 40.Which of these numbers is divisible by 11? (a) 849 (b) 873 (c) 858 (d) 859

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