Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 229 · 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 remainder when 122 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
2. 2.What is the units digit of 7^24? (a) 1 (b) 3 (c) 7 (d) 9
3. 3.Which of these numbers is divisible by 8? (a) 6806 (b) 6810 (c) 6795 (d) 6800
4. 4.What is the smallest number you must add to 209 to make it divisible by 10? (a) 9 (b) 3 (c) 2 (d) 1
5. 5.What is the remainder when 148 is divided by 3? (a) 0 (b) 2 (c) 1
6. 6.What is the smallest number you must add to 370 to make it divisible by 12? (a) 2 (b) 3 (c) 1 (d) 10
7. 7.What is the units digit of 9^52? (a) 2 (b) 9 (c) 1 (d) 0
8. 8.What is the smallest number you must add to 852 to make it divisible by 11? (a) 7 (b) 6 (c) 8 (d) 5
9. 9.Which of these numbers is divisible by 11? (a) 2146 (b) 2176 (c) 2164 (d) 2156
10. 10.What is the remainder when 167 is divided by 7? (a) 4 (b) 1 (c) 6 (d) 5
11. 11.What is the units digit of 2^59? (a) 4 (b) 8 (c) 2 (d) 6
12. 12.Which of these numbers is divisible by 9? (a) 5532 (b) 5526 (c) 5518 (d) 5538
13. 13.What is the remainder when 453 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 5
14. 14.Which of these numbers is divisible by 9? (a) 4835 (b) 4833 (c) 4827 (d) 4850
15. 15.What is the smallest number you must add to 235 to make it divisible by 3? (a) 2 (b) 1
16. 16.What is the smallest number you must add to 227 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
17. 17.Which of these numbers is divisible by 6? (a) 4865 (b) 4868 (c) 4858 (d) 4860
18. 18.Which of these numbers is divisible by 11? (a) 9749 (b) 9735 (c) 9729 (d) 9739
19. 19.Which of these numbers is divisible by 9? (a) 1517 (b) 1522 (c) 1521 (d) 1532
20. 20.Which of these numbers is divisible by 3? (a) 1042 (b) 1046 (c) 1044 (d) 1048
21. 21.What is the remainder when 58 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
22. 22.Which of these numbers is divisible by 8? (a) 6849 (b) 6848 (c) 6846 (d) 6859
23. 23.What is the smallest number you must add to 102 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
24. 24.What is the smallest number you must add to 359 to make it divisible by 9? (a) 3 (b) 2 (c) 1 (d) 8
25. 25.Which of these numbers is divisible by 3? (a) 1888 (b) 1891 (c) 1886 (d) 1887
26. 26.What is the units digit of 9^50? (a) 9 (b) 0 (c) 1 (d) 2
27. 27.What is the smallest number you must add to 145 to make it divisible by 3? (a) 1 (b) 2
28. 28.What is the smallest number you must add to 218 to make it divisible by 3? (a) 2 (b) 1
29. 29.What is the units digit of 9^36? (a) 1 (b) 0 (c) 2 (d) 9
30. 30.What is the smallest number you must add to 276 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
31. 31.What is the remainder when 298 is divided by 10? (a) 9 (b) 8 (c) 2 (d) 7
32. 32.Which of these numbers is divisible by 4? (a) 3414 (b) 3421 (c) 3416 (d) 3418
33. 33.What is the units digit of 8^57? (a) 6 (b) 2 (c) 8 (d) 4
34. 34.What is the units digit of 3^39? (a) 1 (b) 3 (c) 7 (d) 9
35. 35.What is the units digit of 7^45? (a) 7 (b) 1 (c) 3 (d) 9
36. 36.What is the remainder when 482 is divided by 11? (a) 9 (b) 8 (c) 2 (d) 10
37. 37.What is the remainder when 295 is divided by 6? (a) 0 (b) 5 (c) 2 (d) 1
38. 38.What is the remainder when 69 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
39. 39.What is the units digit of 9^56? (a) 9 (b) 1 (c) 0 (d) 2
40. 40.What is the remainder when 435 is divided by 10? (a) 4 (b) 6 (c) 5 (d) 7

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