Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 829 · 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) 6737 (b) 6730 (c) 6736 (d) 6746
2. 2.What is the smallest number you must add to 704 to make it divisible by 9? (a) 7 (b) 8 (c) 2 (d) 6
3. 3.Which of these numbers is divisible by 9? (a) 984 (b) 978 (c) 998 (d) 981
4. 4.What is the smallest number you must add to 172 to make it divisible by 3? (a) 1 (b) 2
5. 5.What is the units digit of 3^47? (a) 3 (b) 1 (c) 9 (d) 7
6. 6.What is the smallest number you must add to 159 to make it divisible by 4? (a) 3 (b) 1 (c) 2
7. 7.Which of these numbers is divisible by 3? (a) 2084 (b) 2085 (c) 2090 (d) 2087
8. 8.What is the remainder when 105 is divided by 8? (a) 7 (b) 2 (c) 0 (d) 1
9. 9.What is the remainder when 604 is divided by 10? (a) 6 (b) 3 (c) 5 (d) 4
10. 10.What is the remainder when 176 is divided by 3? (a) 0 (b) 2 (c) 1
11. 11.Which of these numbers is divisible by 8? (a) 1800 (b) 1801 (c) 1799 (d) 1814
12. 12.What is the smallest number you must add to 124 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
13. 13.Which of these numbers is divisible by 3? (a) 1458 (b) 1456 (c) 1462 (d) 1460
14. 14.What is the smallest number you must add to 306 to make it divisible by 4? (a) 3 (b) 2 (c) 1
15. 15.What is the units digit of 2^17? (a) 4 (b) 2 (c) 6 (d) 8
16. 16.What is the remainder when 69 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
17. 17.Which of these numbers is divisible by 3? (a) 1009 (b) 1003 (c) 1006 (d) 1005
18. 18.What is the remainder when 201 is divided by 5? (a) 4 (b) 0 (c) 2 (d) 1
19. 19.What is the remainder when 448 is divided by 9? (a) 7 (b) 2 (c) 8 (d) 6
20. 20.Which of these numbers is divisible by 8? (a) 6661 (b) 6648 (c) 6653 (d) 6644
21. 21.Which of these numbers is divisible by 6? (a) 4715 (b) 4725 (c) 4719 (d) 4716
22. 22.Which of these numbers is divisible by 6? (a) 5135 (b) 5128 (c) 5130 (d) 5138
23. 23.What is the units digit of 2^22? (a) 2 (b) 4 (c) 6 (d) 8
24. 24.What is the units digit of 7^22? (a) 1 (b) 3 (c) 7 (d) 9
25. 25.What is the remainder when 793 is divided by 12? (a) 2 (b) 11 (c) 1 (d) 0
26. 26.What is the units digit of 9^13? (a) 7 (b) 1 (c) 8 (d) 9
27. 27.What is the units digit of 3^21? (a) 3 (b) 9 (c) 7 (d) 1
28. 28.What is the smallest number you must add to 339 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
29. 29.What is the units digit of 3^54? (a) 3 (b) 1 (c) 7 (d) 9
30. 30.What is the units digit of 7^29? (a) 1 (b) 7 (c) 3 (d) 9
31. 31.Which of these numbers is divisible by 4? (a) 1264 (b) 1270 (c) 1261 (d) 1265
32. 32.Which of these numbers is divisible by 11? (a) 5543 (b) 5559 (c) 5547 (d) 5544
33. 33.What is the smallest number you must add to 354 to make it divisible by 7? (a) 3 (b) 2 (c) 4 (d) 5
34. 34.What is the units digit of 3^48? (a) 3 (b) 1 (c) 9 (d) 7
35. 35.Which of these numbers is divisible by 8? (a) 6000 (b) 6001 (c) 6012 (d) 5999
36. 36.What is the units digit of 4^15? (a) 3 (b) 5 (c) 4 (d) 6
37. 37.Which of these numbers is divisible by 4? (a) 3216 (b) 3213 (c) 3217 (d) 3223
38. 38.What is the units digit of 2^35? (a) 4 (b) 2 (c) 6 (d) 8
39. 39.What is the remainder when 469 is divided by 12? (a) 11 (b) 2 (c) 0 (d) 1
40. 40.Which of these numbers is divisible by 6? (a) 3114 (b) 3124 (c) 3112 (d) 3116

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