Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 95 · 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 6? (a) 2164 (b) 2159 (c) 2160 (d) 2170
2. 2.What is the units digit of 4^22? (a) 4 (b) 7 (c) 6 (d) 5
3. 3.What is the smallest number you must add to 467 to make it divisible by 7? (a) 1 (b) 2 (c) 5 (d) 3
4. 4.What is the remainder when 103 is divided by 6? (a) 5 (b) 0 (c) 1 (d) 2
5. 5.What is the units digit of 2^30? (a) 8 (b) 6 (c) 2 (d) 4
6. 6.Which of these numbers is divisible by 6? (a) 2796 (b) 2803 (c) 2799 (d) 2792
7. 7.What is the remainder when 506 is divided by 9? (a) 3 (b) 2 (c) 7 (d) 1
8. 8.Which of these numbers is divisible by 9? (a) 4976 (b) 4964 (c) 4968 (d) 4979
9. 9.What is the remainder when 366 is divided by 7? (a) 3 (b) 5 (c) 1 (d) 2
10. 10.What is the smallest number you must add to 159 to make it divisible by 9? (a) 3 (b) 6 (c) 4 (d) 2
11. 11.What is the smallest number you must add to 133 to make it divisible by 3? (a) 2 (b) 1
12. 12.What is the remainder when 149 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
13. 13.What is the units digit of 2^50? (a) 4 (b) 6 (c) 8 (d) 2
14. 14.What is the smallest number you must add to 85 to make it divisible by 7? (a) 5 (b) 1 (c) 4 (d) 6
15. 15.What is the units digit of 9^29? (a) 7 (b) 8 (c) 1 (d) 9
16. 16.What is the remainder when 426 is divided by 9? (a) 4 (b) 6 (c) 3 (d) 2
17. 17.Which of these numbers is divisible by 6? (a) 2175 (b) 2165 (c) 2166 (d) 2168
18. 18.What is the smallest number you must add to 131 to make it divisible by 12? (a) 3 (b) 11 (c) 1 (d) 2
19. 19.What is the remainder when 55 is divided by 3? (a) 2 (b) 1 (c) 0
20. 20.What is the units digit of 9^15? (a) 1 (b) 8 (c) 7 (d) 9
21. 21.What is the units digit of 3^53? (a) 1 (b) 7 (c) 9 (d) 3
22. 22.What is the remainder when 249 is divided by 7? (a) 5 (b) 3 (c) 6 (d) 4
23. 23.What is the smallest number you must add to 100 to make it divisible by 3? (a) 2 (b) 1
24. 24.Which of these numbers is divisible by 3? (a) 2279 (b) 2274 (c) 2275 (d) 2272
25. 25.What is the units digit of 4^14? (a) 4 (b) 6 (c) 5 (d) 7
26. 26.Which of these numbers is divisible by 3? (a) 1058 (b) 1061 (c) 1055 (d) 1056
27. 27.Which of these numbers is divisible by 9? (a) 736 (b) 729 (c) 721 (d) 745
28. 28.Which of these numbers is divisible by 11? (a) 6589 (b) 6607 (c) 6588 (d) 6595
29. 29.What is the remainder when 222 is divided by 10? (a) 1 (b) 3 (c) 2 (d) 8
30. 30.What is the remainder when 359 is divided by 10? (a) 1 (b) 9 (c) 8 (d) 7
31. 31.What is the remainder when 210 is divided by 8? (a) 2 (b) 6 (c) 3 (d) 1
32. 32.What is the smallest number you must add to 573 to make it divisible by 8? (a) 5 (b) 4 (c) 3 (d) 2
33. 33.What is the smallest number you must add to 355 to make it divisible by 10? (a) 6 (b) 4 (c) 7 (d) 5
34. 34.What is the remainder when 459 is divided by 6? (a) 3 (b) 2 (c) 4 (d) 5
35. 35.What is the remainder when 692 is divided by 10? (a) 1 (b) 2 (c) 3 (d) 8
36. 36.Which of these numbers is divisible by 3? (a) 1359 (b) 1364 (c) 1361 (d) 1357
37. 37.What is the smallest number you must add to 609 to make it divisible by 10? (a) 2 (b) 1 (c) 9 (d) 3
38. 38.Which of these numbers is divisible by 3? (a) 1917 (b) 1918 (c) 1921 (d) 1916
39. 39.What is the units digit of 3^54? (a) 9 (b) 1 (c) 7 (d) 3
40. 40.What is the remainder when 70 is divided by 3? (a) 0 (b) 1 (c) 2

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