Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 366 · 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 4? (a) 1143 (b) 1146 (c) 1144 (d) 1150
2. 2.Which of these numbers is divisible by 8? (a) 1904 (b) 1908 (c) 1919 (d) 1899
3. 3.What is the remainder when 101 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
4. 4.Which of these numbers is divisible by 3? (a) 2046 (b) 2050 (c) 2044 (d) 2048
5. 5.Which of these numbers is divisible by 3? (a) 1763 (b) 1764 (c) 1766 (d) 1769
6. 6.Which of these numbers is divisible by 9? (a) 1151 (b) 1134 (c) 1133 (d) 1138
7. 7.What is the smallest number you must add to 242 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 2
8. 8.Which of these numbers is divisible by 9? (a) 2367 (b) 2383 (c) 2374 (d) 2364
9. 9.What is the units digit of 7^40? (a) 9 (b) 3 (c) 7 (d) 1
10. 10.Which of these numbers is divisible by 11? (a) 2965 (b) 2950 (c) 2948 (d) 2947
11. 11.What is the smallest number you must add to 116 to make it divisible by 8? (a) 4 (b) 3 (c) 6 (d) 5
12. 12.Which of these numbers is divisible by 4? (a) 789 (b) 792 (c) 798 (d) 793
13. 13.Which of these numbers is divisible by 3? (a) 2539 (b) 2541 (c) 2543 (d) 2545
14. 14.Which of these numbers is divisible by 9? (a) 3269 (b) 3258 (c) 3255 (d) 3266
15. 15.What is the remainder when 203 is divided by 6? (a) 3 (b) 5 (c) 4 (d) 1
16. 16.What is the units digit of 4^31? (a) 6 (b) 5 (c) 3 (d) 4
17. 17.What is the units digit of 9^18? (a) 9 (b) 0 (c) 2 (d) 1
18. 18.Which of these numbers is divisible by 9? (a) 5098 (b) 5114 (c) 5103 (d) 5110
19. 19.Which of these numbers is divisible by 8? (a) 7058 (b) 7054 (c) 7067 (d) 7056
20. 20.Which of these numbers is divisible by 6? (a) 2246 (b) 2261 (c) 2251 (d) 2250
21. 21.Which of these numbers is divisible by 11? (a) 8261 (b) 8260 (c) 8271 (d) 8279
22. 22.What is the remainder when 162 is divided by 10? (a) 8 (b) 3 (c) 2 (d) 1
23. 23.What is the units digit of 8^57? (a) 8 (b) 4 (c) 6 (d) 2
24. 24.What is the remainder when 394 is divided by 9? (a) 2 (b) 6 (c) 7 (d) 8
25. 25.What is the units digit of 8^14? (a) 4 (b) 2 (c) 6 (d) 8
26. 26.What is the smallest number you must add to 459 to make it divisible by 7? (a) 4 (b) 5 (c) 2 (d) 3
27. 27.What is the units digit of 7^57? (a) 1 (b) 3 (c) 9 (d) 7
28. 28.What is the smallest number you must add to 221 to make it divisible by 11? (a) 10 (b) 9 (c) 1 (d) 8
29. 29.What is the units digit of 2^14? (a) 2 (b) 6 (c) 4 (d) 8
30. 30.What is the smallest number you must add to 289 to make it divisible by 9? (a) 1 (b) 8 (c) 6 (d) 7
31. 31.Which of these numbers is divisible by 8? (a) 7073 (b) 7070 (c) 7062 (d) 7064
32. 32.What is the smallest number you must add to 62 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
33. 33.What is the smallest number you must add to 742 to make it divisible by 10? (a) 9 (b) 8 (c) 7 (d) 2
34. 34.Which of these numbers is divisible by 9? (a) 1372 (b) 1359 (c) 1361 (d) 1354
35. 35.What is the units digit of 4^20? (a) 5 (b) 7 (c) 6 (d) 4
36. 36.What is the remainder when 260 is divided by 9? (a) 6 (b) 8 (c) 7 (d) 1
37. 37.What is the units digit of 3^50? (a) 7 (b) 1 (c) 3 (d) 9
38. 38.Which of these numbers is divisible by 4? (a) 2851 (b) 2845 (c) 2844 (d) 2842
39. 39.Which of these numbers is divisible by 9? (a) 3052 (b) 3061 (c) 3060 (d) 3070
40. 40.What is the smallest number you must add to 349 to make it divisible by 7? (a) 3 (b) 1 (c) 6 (d) 2

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