Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 78 · 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 units digit of 3^15? (a) 7 (b) 1 (c) 9 (d) 3
2. 2.What is the smallest number you must add to 295 to make it divisible by 8? (a) 2 (b) 7 (c) 3 (d) 1
3. 3.What is the remainder when 326 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 0
4. 4.Which of these numbers is divisible by 9? (a) 916 (b) 919 (c) 909 (d) 905
5. 5.What is the units digit of 9^13? (a) 7 (b) 8 (c) 1 (d) 9
6. 6.What is the remainder when 241 is divided by 9? (a) 6 (b) 2 (c) 8 (d) 7
7. 7.Which of these numbers is divisible by 8? (a) 7187 (b) 7192 (c) 7203 (d) 7197
8. 8.What is the smallest number you must add to 298 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
9. 9.What is the smallest number you must add to 118 to make it divisible by 4? (a) 2 (b) 3 (c) 1
10. 10.What is the units digit of 9^27? (a) 9 (b) 7 (c) 1 (d) 8
11. 11.What is the smallest number you must add to 386 to make it divisible by 6? (a) 4 (b) 3 (c) 5 (d) 2
12. 12.What is the remainder when 141 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
13. 13.What is the units digit of 8^13? (a) 6 (b) 8 (c) 4 (d) 2
14. 14.Which of these numbers is divisible by 3? (a) 1060 (b) 1062 (c) 1063 (d) 1067
15. 15.What is the smallest number you must add to 634 to make it divisible by 8? (a) 5 (b) 6 (c) 2 (d) 7
16. 16.Which of these numbers is divisible by 11? (a) 3388 (b) 3390 (c) 3405 (d) 3378
17. 17.Which of these numbers is divisible by 3? (a) 963 (b) 965 (c) 968 (d) 962
18. 18.What is the remainder when 184 is divided by 9? (a) 3 (b) 5 (c) 6 (d) 4
19. 19.What is the units digit of 4^10? (a) 5 (b) 4 (c) 6 (d) 7
20. 20.What is the remainder when 206 is divided by 3? (a) 0 (b) 1 (c) 2
21. 21.What is the smallest number you must add to 307 to make it divisible by 9? (a) 6 (b) 8 (c) 1 (d) 7
22. 22.What is the remainder when 772 is divided by 10? (a) 1 (b) 3 (c) 8 (d) 2
23. 23.What is the remainder when 395 is divided by 7? (a) 2 (b) 3 (c) 5 (d) 4
24. 24.What is the remainder when 130 is divided by 3? (a) 2 (b) 1 (c) 0
25. 25.What is the units digit of 4^53? (a) 5 (b) 6 (c) 3 (d) 4
26. 26.What is the remainder when 159 is divided by 7? (a) 5 (b) 2 (c) 6 (d) 4
27. 27.What is the smallest number you must add to 134 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
28. 28.What is the smallest number you must add to 285 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 4
29. 29.What is the units digit of 8^30? (a) 8 (b) 6 (c) 4 (d) 2
30. 30.Which of these numbers is divisible by 3? (a) 898 (b) 896 (c) 901 (d) 897
31. 31.Which of these numbers is divisible by 9? (a) 1787 (b) 1775 (c) 1782 (d) 1794
32. 32.What is the units digit of 7^25? (a) 3 (b) 1 (c) 9 (d) 7
33. 33.What is the remainder when 227 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
34. 34.What is the smallest number you must add to 157 to make it divisible by 11? (a) 3 (b) 9 (c) 8 (d) 7
35. 35.Which of these numbers is divisible by 3? (a) 1811 (b) 1817 (c) 1812 (d) 1814
36. 36.What is the smallest number you must add to 302 to make it divisible by 10? (a) 7 (b) 8 (c) 2 (d) 9
37. 37.What is the units digit of 9^14? (a) 1 (b) 2 (c) 9 (d) 0
38. 38.Which of these numbers is divisible by 6? (a) 2538 (b) 2535 (c) 2545 (d) 2539
39. 39.What is the smallest number you must add to 207 to make it divisible by 7? (a) 3 (b) 4 (c) 2 (d) 5
40. 40.What is the units digit of 8^22? (a) 4 (b) 2 (c) 6 (d) 8

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