Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 876 · 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.Which of these numbers is divisible by 11? (a) 5373 (b) 5379 (c) 5385 (d) 5399
2. 2.What is the units digit of 4^55? (a) 3 (b) 4 (c) 5 (d) 6
3. 3.What is the remainder when 429 is divided by 6? (a) 5 (b) 2 (c) 3 (d) 4
4. 4.What is the smallest number you must add to 134 to make it divisible by 9? (a) 3 (b) 8 (c) 2 (d) 1
5. 5.Which of these numbers is divisible by 8? (a) 3912 (b) 3909 (c) 3913 (d) 3921
6. 6.What is the smallest number you must add to 293 to make it divisible by 8? (a) 3 (b) 5 (c) 4 (d) 2
7. 7.What is the smallest number you must add to 128 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
8. 8.Which of these numbers is divisible by 4? (a) 1298 (b) 1292 (c) 1289 (d) 1293
9. 9.What is the smallest number you must add to 449 to make it divisible by 11? (a) 3 (b) 1 (c) 2 (d) 9
10. 10.What is the remainder when 466 is divided by 8? (a) 2 (b) 3 (c) 6 (d) 1
11. 11.What is the smallest number you must add to 450 to make it divisible by 8? (a) 6 (b) 7 (c) 2 (d) 5
12. 12.What is the remainder when 197 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
13. 13.What is the units digit of 9^53? (a) 7 (b) 9 (c) 8 (d) 1
14. 14.What is the remainder when 164 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 2
15. 15.What is the units digit of 3^45? (a) 7 (b) 3 (c) 1 (d) 9
16. 16.What is the remainder when 501 is divided by 10? (a) 0 (b) 2 (c) 1 (d) 9
17. 17.Which of these numbers is divisible by 3? (a) 2394 (b) 2399 (c) 2393 (d) 2395
18. 18.Which of these numbers is divisible by 3? (a) 369 (b) 368 (c) 374 (d) 370
19. 19.What is the units digit of 8^60? (a) 4 (b) 8 (c) 2 (d) 6
20. 20.What is the units digit of 9^46? (a) 0 (b) 1 (c) 2 (d) 9
21. 21.What is the units digit of 8^47? (a) 6 (b) 8 (c) 4 (d) 2
22. 22.Which of these numbers is divisible by 4? (a) 180 (b) 178 (c) 181 (d) 187
23. 23.Which of these numbers is divisible by 6? (a) 842 (b) 834 (c) 831 (d) 838
24. 24.What is the remainder when 427 is divided by 11? (a) 2 (b) 10 (c) 8 (d) 9
25. 25.What is the remainder when 145 is divided by 3? (a) 2 (b) 0 (c) 1
26. 26.Which of these numbers is divisible by 4? (a) 2707 (b) 2697 (c) 2701 (d) 2700
27. 27.Which of these numbers is divisible by 4? (a) 3432 (b) 3435 (c) 3431 (d) 3439
28. 28.What is the remainder when 89 is divided by 3? (a) 1 (b) 2 (c) 0
29. 29.Which of these numbers is divisible by 4? (a) 1807 (b) 1802 (c) 1809 (d) 1804
30. 30.What is the remainder when 66 is divided by 5? (a) 4 (b) 2 (c) 0 (d) 1
31. 31.What is the units digit of 4^17? (a) 5 (b) 3 (c) 6 (d) 4
32. 32.What is the smallest number you must add to 159 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 4
33. 33.Which of these numbers is divisible by 11? (a) 1744 (b) 1749 (c) 1762 (d) 1751
34. 34.Which of these numbers is divisible by 3? (a) 2111 (b) 2109 (c) 2107 (d) 2113
35. 35.Which of these numbers is divisible by 3? (a) 1187 (b) 1189 (c) 1184 (d) 1185
36. 36.What is the units digit of 9^52? (a) 9 (b) 0 (c) 1 (d) 2
37. 37.What is the units digit of 9^20? (a) 0 (b) 9 (c) 2 (d) 1
38. 38.Which of these numbers is divisible by 11? (a) 9529 (b) 9538 (c) 9537 (d) 9549
39. 39.What is the smallest number you must add to 417 to make it divisible by 9? (a) 6 (b) 7 (c) 5 (d) 3
40. 40.Which of these numbers is divisible by 4? (a) 324 (b) 323 (c) 330 (d) 326

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