Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 854 · 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 8^45? (a) 2 (b) 4 (c) 8 (d) 6
2. 2.Which of these numbers is divisible by 9? (a) 1407 (b) 1413 (c) 1416 (d) 1429
3. 3.Which of these numbers is divisible by 3? (a) 149 (b) 152 (c) 145 (d) 147
4. 4.What is the units digit of 3^20? (a) 3 (b) 7 (c) 1 (d) 9
5. 5.What is the remainder when 114 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
6. 6.What is the remainder when 239 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
7. 7.What is the smallest number you must add to 361 to make it divisible by 10? (a) 1 (b) 8 (c) 9 (d) 7
8. 8.What is the remainder when 585 is divided by 11? (a) 1 (b) 9 (c) 2 (d) 3
9. 9.Which of these numbers is divisible by 6? (a) 1092 (b) 1100 (c) 1089 (d) 1093
10. 10.Which of these numbers is divisible by 9? (a) 5508 (b) 5502 (c) 5513 (d) 5518
11. 11.What is the units digit of 3^48? (a) 3 (b) 1 (c) 9 (d) 7
12. 12.What is the remainder when 201 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 0
13. 13.Which of these numbers is divisible by 6? (a) 1974 (b) 1982 (c) 1977 (d) 1973
14. 14.What is the units digit of 7^39? (a) 3 (b) 7 (c) 9 (d) 1
15. 15.Which of these numbers is divisible by 3? (a) 132 (b) 133 (c) 130 (d) 136
16. 16.Which of these numbers is divisible by 3? (a) 1766 (b) 1759 (c) 1762 (d) 1761
17. 17.What is the units digit of 9^46? (a) 0 (b) 2 (c) 9 (d) 1
18. 18.What is the remainder when 217 is divided by 3? (a) 0 (b) 1 (c) 2
19. 19.Which of these numbers is divisible by 6? (a) 2099 (b) 2091 (c) 2087 (d) 2088
20. 20.What is the remainder when 100 is divided by 6? (a) 2 (b) 4 (c) 5 (d) 3
21. 21.What is the smallest number you must add to 41 to make it divisible by 4? (a) 2 (b) 1 (c) 3
22. 22.What is the smallest number you must add to 295 to make it divisible by 6? (a) 4 (b) 1 (c) 3 (d) 5
23. 23.What is the units digit of 4^33? (a) 6 (b) 5 (c) 3 (d) 4
24. 24.What is the units digit of 4^50? (a) 4 (b) 6 (c) 5 (d) 7
25. 25.What is the smallest number you must add to 78 to make it divisible by 7? (a) 1 (b) 4 (c) 5 (d) 6
26. 26.What is the remainder when 491 is divided by 11? (a) 4 (b) 6 (c) 8 (d) 7
27. 27.What is the smallest number you must add to 116 to make it divisible by 8? (a) 5 (b) 4 (c) 3 (d) 6
28. 28.What is the smallest number you must add to 124 to make it divisible by 10? (a) 5 (b) 6 (c) 4 (d) 7
29. 29.What is the units digit of 7^52? (a) 9 (b) 3 (c) 7 (d) 1
30. 30.Which of these numbers is divisible by 11? (a) 7014 (b) 7027 (c) 7036 (d) 7018
31. 31.What is the smallest number you must add to 255 to make it divisible by 8? (a) 3 (b) 7 (c) 2 (d) 1
32. 32.Which of these numbers is divisible by 6? (a) 1648 (b) 1638 (c) 1635 (d) 1641
33. 33.What is the smallest number you must add to 460 to make it divisible by 6? (a) 3 (b) 4 (c) 2 (d) 1
34. 34.Which of these numbers is divisible by 3? (a) 193 (b) 189 (c) 187 (d) 190
35. 35.Which of these numbers is divisible by 4? (a) 3443 (b) 3436 (c) 3437 (d) 3435
36. 36.What is the remainder when 162 is divided by 8? (a) 1 (b) 2 (c) 6 (d) 3
37. 37.What is the remainder when 531 is divided by 10? (a) 2 (b) 1 (c) 0 (d) 9
38. 38.Which of these numbers is divisible by 4? (a) 689 (b) 693 (c) 685 (d) 688
39. 39.What is the remainder when 285 is divided by 7? (a) 5 (b) 6 (c) 4 (d) 2
40. 40.Which of these numbers is divisible by 6? (a) 2857 (b) 2862 (c) 2865 (d) 2869

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