Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 787 · 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 314 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
2. 2.Which of these numbers is divisible by 4? (a) 3386 (b) 3384 (c) 3389 (d) 3382
3. 3.What is the remainder when 214 is divided by 10? (a) 4 (b) 6 (c) 3 (d) 5
4. 4.What is the remainder when 106 is divided by 5? (a) 1 (b) 2 (c) 0 (d) 4
5. 5.Which of these numbers is divisible by 11? (a) 8185 (b) 8184 (c) 8198 (d) 8179
6. 6.What is the remainder when 805 is divided by 10? (a) 6 (b) 5 (c) 7 (d) 4
7. 7.What is the remainder when 656 is divided by 10? (a) 7 (b) 4 (c) 5 (d) 6
8. 8.What is the smallest number you must add to 241 to make it divisible by 3? (a) 1 (b) 2
9. 9.What is the remainder when 266 is divided by 6? (a) 4 (b) 3 (c) 1 (d) 2
10. 10.What is the remainder when 523 is divided by 12? (a) 7 (b) 6 (c) 5 (d) 8
11. 11.What is the smallest number you must add to 115 to make it divisible by 3? (a) 1 (b) 2
12. 12.What is the smallest number you must add to 293 to make it divisible by 4? (a) 2 (b) 1 (c) 3
13. 13.What is the units digit of 9^18? (a) 2 (b) 0 (c) 1 (d) 9
14. 14.What is the remainder when 346 is divided by 6? (a) 2 (b) 5 (c) 3 (d) 4
15. 15.Which of these numbers is divisible by 11? (a) 6530 (b) 6549 (c) 6535 (d) 6534
16. 16.What is the smallest number you must add to 443 to make it divisible by 8? (a) 6 (b) 3 (c) 5 (d) 4
17. 17.What is the remainder when 309 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
18. 18.What is the smallest number you must add to 187 to make it divisible by 3? (a) 2 (b) 1
19. 19.What is the smallest number you must add to 142 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
20. 20.Which of these numbers is divisible by 3? (a) 2495 (b) 2492 (c) 2497 (d) 2493
21. 21.What is the smallest number you must add to 44 to make it divisible by 3? (a) 2 (b) 1
22. 22.What is the remainder when 336 is divided by 5? (a) 0 (b) 2 (c) 1 (d) 4
23. 23.What is the smallest number you must add to 62 to make it divisible by 3? (a) 1 (b) 2
24. 24.What is the smallest number you must add to 488 to make it divisible by 9? (a) 8 (b) 2 (c) 7 (d) 6
25. 25.What is the units digit of 8^52? (a) 8 (b) 2 (c) 6 (d) 4
26. 26.What is the remainder when 285 is divided by 12? (a) 8 (b) 9 (c) 10 (d) 3
27. 27.Which of these numbers is divisible by 3? (a) 251 (b) 245 (c) 246 (d) 248
28. 28.What is the remainder when 274 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
29. 29.What is the remainder when 391 is divided by 10? (a) 0 (b) 9 (c) 1 (d) 2
30. 30.What is the units digit of 7^14? (a) 3 (b) 7 (c) 9 (d) 1
31. 31.Which of these numbers is divisible by 6? (a) 807 (b) 796 (c) 800 (d) 798
32. 32.What is the remainder when 734 is divided by 10? (a) 4 (b) 3 (c) 5 (d) 6
33. 33.What is the remainder when 136 is divided by 7? (a) 4 (b) 2 (c) 5 (d) 3
34. 34.What is the remainder when 213 is divided by 8? (a) 5 (b) 4 (c) 3 (d) 6
35. 35.What is the units digit of 9^42? (a) 9 (b) 0 (c) 1 (d) 2
36. 36.Which of these numbers is divisible by 4? (a) 1114 (b) 1109 (c) 1117 (d) 1112
37. 37.Which of these numbers is divisible by 8? (a) 4706 (b) 4715 (c) 4702 (d) 4704
38. 38.Which of these numbers is divisible by 11? (a) 1199 (b) 1205 (c) 1193 (d) 1212
39. 39.What is the smallest number you must add to 326 to make it divisible by 7? (a) 4 (b) 5 (c) 2 (d) 3
40. 40.What is the units digit of 7^17? (a) 3 (b) 1 (c) 7 (d) 9

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