Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 560 · 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 422 is divided by 8? (a) 5 (b) 7 (c) 2 (d) 6
2. 2.What is the units digit of 9^58? (a) 9 (b) 0 (c) 2 (d) 1
3. 3.What is the smallest number you must add to 222 to make it divisible by 9? (a) 3 (b) 6 (c) 2 (d) 4
4. 4.What is the remainder when 267 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
5. 5.Which of these numbers is divisible by 8? (a) 2854 (b) 2866 (c) 2856 (d) 2862
6. 6.What is the smallest number you must add to 186 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
7. 7.What is the units digit of 3^55? (a) 1 (b) 3 (c) 9 (d) 7
8. 8.What is the units digit of 7^19? (a) 9 (b) 3 (c) 1 (d) 7
9. 9.Which of these numbers is divisible by 3? (a) 376 (b) 370 (c) 372 (d) 373
10. 10.What is the units digit of 4^53? (a) 5 (b) 3 (c) 6 (d) 4
11. 11.Which of these numbers is divisible by 9? (a) 7301 (b) 7313 (c) 7321 (d) 7308
12. 12.Which of these numbers is divisible by 9? (a) 5183 (b) 5188 (c) 5170 (d) 5175
13. 13.What is the units digit of 8^49? (a) 6 (b) 8 (c) 2 (d) 4
14. 14.Which of these numbers is divisible by 4? (a) 890 (b) 895 (c) 892 (d) 899
15. 15.Which of these numbers is divisible by 11? (a) 2919 (b) 2926 (c) 2945 (d) 2931
16. 16.What is the smallest number you must add to 31 to make it divisible by 3? (a) 2 (b) 1
17. 17.What is the remainder when 217 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
18. 18.What is the units digit of 8^40? (a) 6 (b) 2 (c) 8 (d) 4
19. 19.What is the remainder when 564 is divided by 8? (a) 5 (b) 6 (c) 4 (d) 3
20. 20.What is the units digit of 3^54? (a) 7 (b) 3 (c) 1 (d) 9
21. 21.Which of these numbers is divisible by 6? (a) 1633 (b) 1641 (c) 1627 (d) 1632
22. 22.What is the smallest number you must add to 269 to make it divisible by 4? (a) 1 (b) 3 (c) 2
23. 23.Which of these numbers is divisible by 3? (a) 535 (b) 534 (c) 539 (d) 533
24. 24.Which of these numbers is divisible by 9? (a) 1894 (b) 1880 (c) 1881 (d) 1882
25. 25.What is the units digit of 3^25? (a) 3 (b) 7 (c) 9 (d) 1
26. 26.What is the smallest number you must add to 199 to make it divisible by 3? (a) 1 (b) 2
27. 27.What is the remainder when 288 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
28. 28.Which of these numbers is divisible by 6? (a) 2907 (b) 2914 (c) 2904 (d) 2900
29. 29.What is the units digit of 9^14? (a) 2 (b) 0 (c) 1 (d) 9
30. 30.What is the remainder when 673 is divided by 11? (a) 9 (b) 1 (c) 3 (d) 2
31. 31.What is the smallest number you must add to 467 to make it divisible by 7? (a) 5 (b) 3 (c) 1 (d) 2
32. 32.Which of these numbers is divisible by 6? (a) 3461 (b) 3456 (c) 3466 (d) 3455
33. 33.Which of these numbers is divisible by 4? (a) 1866 (b) 1860 (c) 1858 (d) 1862
34. 34.Which of these numbers is divisible by 6? (a) 3499 (b) 3491 (c) 3493 (d) 3492
35. 35.What is the smallest number you must add to 311 to make it divisible by 6? (a) 3 (b) 5 (c) 1 (d) 2
36. 36.Which of these numbers is divisible by 9? (a) 7285 (b) 7290 (c) 7301 (d) 7291
37. 37.What is the remainder when 214 is divided by 3? (a) 2 (b) 0 (c) 1
38. 38.What is the units digit of 9^11? (a) 1 (b) 8 (c) 9 (d) 7
39. 39.Which of these numbers is divisible by 11? (a) 4201 (b) 4209 (c) 4181 (d) 4191
40. 40.Which of these numbers is divisible by 9? (a) 4985 (b) 4986 (c) 4992 (d) 5001

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