Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 819 · 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 8? (a) 763 (b) 781 (c) 768 (d) 769
2. 2.What is the units digit of 8^10? (a) 4 (b) 2 (c) 6 (d) 8
3. 3.What is the remainder when 317 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
4. 4.Which of these numbers is divisible by 8? (a) 1047 (b) 1055 (c) 1058 (d) 1048
5. 5.What is the units digit of 7^48? (a) 1 (b) 7 (c) 3 (d) 9
6. 6.Which of these numbers is divisible by 11? (a) 5115 (b) 5121 (c) 5128 (d) 5108
7. 7.What is the units digit of 3^13? (a) 9 (b) 1 (c) 7 (d) 3
8. 8.What is the units digit of 3^10? (a) 3 (b) 7 (c) 9 (d) 1
9. 9.What is the remainder when 58 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
10. 10.What is the units digit of 2^10? (a) 8 (b) 2 (c) 4 (d) 6
11. 11.What is the smallest number you must add to 210 to make it divisible by 8? (a) 2 (b) 7 (c) 5 (d) 6
12. 12.What is the units digit of 8^14? (a) 4 (b) 2 (c) 6 (d) 8
13. 13.What is the smallest number you must add to 163 to make it divisible by 8? (a) 4 (b) 3 (c) 5 (d) 6
14. 14.What is the smallest number you must add to 155 to make it divisible by 12? (a) 2 (b) 3 (c) 11 (d) 1
15. 15.What is the remainder when 279 is divided by 12? (a) 9 (b) 4 (c) 3 (d) 2
16. 16.Which of these numbers is divisible by 11? (a) 9334 (b) 9323 (c) 9314 (d) 9317
17. 17.What is the units digit of 9^53? (a) 9 (b) 1 (c) 7 (d) 8
18. 18.What is the units digit of 3^21? (a) 3 (b) 9 (c) 7 (d) 1
19. 19.Which of these numbers is divisible by 6? (a) 1359 (b) 1351 (c) 1364 (d) 1356
20. 20.Which of these numbers is divisible by 8? (a) 981 (b) 997 (c) 985 (d) 984
21. 21.What is the remainder when 74 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
22. 22.What is the units digit of 2^55? (a) 4 (b) 6 (c) 2 (d) 8
23. 23.Which of these numbers is divisible by 9? (a) 4458 (b) 4455 (c) 4471 (d) 4449
24. 24.Which of these numbers is divisible by 3? (a) 2383 (b) 2386 (c) 2381 (d) 2382
25. 25.Which of these numbers is divisible by 3? (a) 263 (b) 265 (c) 264 (d) 268
26. 26.What is the remainder when 271 is divided by 5? (a) 1 (b) 0 (c) 2 (d) 4
27. 27.Which of these numbers is divisible by 3? (a) 464 (b) 466 (c) 465 (d) 469
28. 28.What is the remainder when 167 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
29. 29.What is the remainder when 211 is divided by 7? (a) 6 (b) 2 (c) 0 (d) 1
30. 30.What is the remainder when 124 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
31. 31.What is the units digit of 8^45? (a) 2 (b) 6 (c) 4 (d) 8
32. 32.What is the units digit of 4^55? (a) 3 (b) 6 (c) 5 (d) 4
33. 33.What is the units digit of 4^22? (a) 5 (b) 4 (c) 6 (d) 7
34. 34.What is the smallest number you must add to 691 to make it divisible by 11? (a) 9 (b) 1 (c) 3 (d) 2
35. 35.Which of these numbers is divisible by 4? (a) 521 (b) 526 (c) 531 (d) 524
36. 36.Which of these numbers is divisible by 6? (a) 2133 (b) 2127 (c) 2139 (d) 2130
37. 37.What is the units digit of 4^56? (a) 4 (b) 6 (c) 7 (d) 5
38. 38.Which of these numbers is divisible by 3? (a) 143 (b) 144 (c) 145 (d) 149
39. 39.What is the remainder when 395 is divided by 9? (a) 1 (b) 6 (c) 8 (d) 7
40. 40.What is the units digit of 8^39? (a) 2 (b) 4 (c) 8 (d) 6

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