Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 801 · 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 8^10? (a) 6 (b) 8 (c) 2 (d) 4
2. 2.Which of these numbers is divisible by 6? (a) 5154 (b) 5155 (c) 5152 (d) 5164
3. 3.What is the smallest number you must add to 123 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
4. 4.What is the units digit of 2^53? (a) 6 (b) 4 (c) 2 (d) 8
5. 5.What is the units digit of 8^36? (a) 2 (b) 8 (c) 6 (d) 4
6. 6.Which of these numbers is divisible by 6? (a) 1482 (b) 1481 (c) 1493 (d) 1485
7. 7.What is the units digit of 3^13? (a) 1 (b) 7 (c) 3 (d) 9
8. 8.Which of these numbers is divisible by 6? (a) 2101 (b) 2099 (c) 2100 (d) 2107
9. 9.What is the smallest number you must add to 110 to make it divisible by 4? (a) 2 (b) 3 (c) 1
10. 10.What is the smallest number you must add to 169 to make it divisible by 9? (a) 3 (b) 1 (c) 7 (d) 2
11. 11.Which of these numbers is divisible by 4? (a) 1958 (b) 1960 (c) 1961 (d) 1965
12. 12.What is the smallest number you must add to 254 to make it divisible by 12? (a) 11 (b) 9 (c) 2 (d) 10
13. 13.What is the smallest number you must add to 506 to make it divisible by 10? (a) 6 (b) 3 (c) 4 (d) 5
14. 14.What is the units digit of 2^22? (a) 8 (b) 4 (c) 6 (d) 2
15. 15.What is the units digit of 7^30? (a) 9 (b) 7 (c) 3 (d) 1
16. 16.What is the remainder when 82 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
17. 17.What is the smallest number you must add to 152 to make it divisible by 6? (a) 5 (b) 3 (c) 2 (d) 4
18. 18.What is the units digit of 3^21? (a) 3 (b) 1 (c) 7 (d) 9
19. 19.What is the remainder when 695 is divided by 9? (a) 2 (b) 3 (c) 1 (d) 7
20. 20.What is the units digit of 2^21? (a) 6 (b) 4 (c) 8 (d) 2
21. 21.Which of these numbers is divisible by 8? (a) 4872 (b) 4879 (c) 4865 (d) 4881
22. 22.Which of these numbers is divisible by 4? (a) 1843 (b) 1838 (c) 1840 (d) 1847
23. 23.What is the smallest number you must add to 388 to make it divisible by 11? (a) 7 (b) 8 (c) 9 (d) 3
24. 24.What is the smallest number you must add to 745 to make it divisible by 11? (a) 4 (b) 2 (c) 3 (d) 8
25. 25.Which of these numbers is divisible by 11? (a) 1199 (b) 1203 (c) 1190 (d) 1213
26. 26.What is the smallest number you must add to 676 to make it divisible by 10? (a) 5 (b) 6 (c) 4 (d) 3
27. 27.What is the remainder when 585 is divided by 8? (a) 1 (b) 2 (c) 0 (d) 7
28. 28.What is the remainder when 49 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
29. 29.What is the units digit of 3^19? (a) 9 (b) 1 (c) 3 (d) 7
30. 30.What is the units digit of 8^11? (a) 8 (b) 2 (c) 6 (d) 4
31. 31.What is the units digit of 4^36? (a) 7 (b) 5 (c) 4 (d) 6
32. 32.What is the units digit of 2^15? (a) 6 (b) 4 (c) 2 (d) 8
33. 33.What is the smallest number you must add to 544 to make it divisible by 12? (a) 9 (b) 7 (c) 4 (d) 8
34. 34.Which of these numbers is divisible by 8? (a) 6026 (b) 6021 (c) 6015 (d) 6016
35. 35.What is the smallest number you must add to 491 to make it divisible by 8? (a) 3 (b) 6 (c) 4 (d) 5
36. 36.What is the units digit of 4^40? (a) 4 (b) 5 (c) 6 (d) 7
37. 37.Which of these numbers is divisible by 6? (a) 1152 (b) 1150 (c) 1159 (d) 1156
38. 38.What is the smallest number you must add to 259 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
39. 39.What is the remainder when 466 is divided by 6? (a) 3 (b) 5 (c) 2 (d) 4
40. 40.What is the remainder when 557 is divided by 9? (a) 1 (b) 7 (c) 8 (d) 6

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