Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 267 · 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 596 is divided by 8? (a) 3 (b) 6 (c) 5 (d) 4
2. 2.Which of these numbers is divisible by 6? (a) 4697 (b) 4686 (c) 4682 (d) 4691
3. 3.What is the units digit of 9^13? (a) 7 (b) 9 (c) 1 (d) 8
4. 4.Which of these numbers is divisible by 9? (a) 2126 (b) 2110 (c) 2115 (d) 2120
5. 5.What is the units digit of 4^32? (a) 4 (b) 5 (c) 7 (d) 6
6. 6.Which of these numbers is divisible by 9? (a) 4003 (b) 3979 (c) 3993 (d) 3987
7. 7.What is the smallest number you must add to 172 to make it divisible by 6? (a) 2 (b) 1 (c) 4 (d) 3
8. 8.What is the units digit of 4^47? (a) 3 (b) 4 (c) 6 (d) 5
9. 9.What is the units digit of 2^16? (a) 2 (b) 8 (c) 4 (d) 6
10. 10.Which of these numbers is divisible by 3? (a) 986 (b) 987 (c) 989 (d) 992
11. 11.What is the smallest number you must add to 258 to make it divisible by 12? (a) 7 (b) 6 (c) 8 (d) 5
12. 12.What is the units digit of 2^57? (a) 6 (b) 2 (c) 8 (d) 4
13. 13.What is the smallest number you must add to 124 to make it divisible by 3? (a) 1 (b) 2
14. 14.What is the remainder when 520 is divided by 9? (a) 2 (b) 8 (c) 7 (d) 6
15. 15.What is the remainder when 548 is divided by 11? (a) 9 (b) 8 (c) 2 (d) 10
16. 16.What is the units digit of 2^53? (a) 2 (b) 8 (c) 4 (d) 6
17. 17.What is the remainder when 181 is divided by 5? (a) 4 (b) 0 (c) 1 (d) 2
18. 18.What is the units digit of 7^20? (a) 1 (b) 9 (c) 7 (d) 3
19. 19.What is the remainder when 177 is divided by 7? (a) 2 (b) 3 (c) 5 (d) 1
20. 20.Which of these numbers is divisible by 4? (a) 511 (b) 504 (c) 506 (d) 501
21. 21.What is the smallest number you must add to 407 to make it divisible by 8? (a) 2 (b) 3 (c) 7 (d) 1
22. 22.Which of these numbers is divisible by 8? (a) 6589 (b) 6580 (c) 6576 (d) 6571
23. 23.Which of these numbers is divisible by 6? (a) 1597 (b) 1602 (c) 1607 (d) 1613
24. 24.What is the units digit of 4^28? (a) 4 (b) 5 (c) 7 (d) 6
25. 25.Which of these numbers is divisible by 11? (a) 4358 (b) 4356 (c) 4351 (d) 4369
26. 26.What is the units digit of 2^10? (a) 2 (b) 6 (c) 4 (d) 8
27. 27.What is the units digit of 7^59? (a) 7 (b) 9 (c) 3 (d) 1
28. 28.Which of these numbers is divisible by 9? (a) 3158 (b) 3161 (c) 3175 (d) 3159
29. 29.Which of these numbers is divisible by 11? (a) 828 (b) 825 (c) 838 (d) 822
30. 30.Which of these numbers is divisible by 4? (a) 987 (b) 984 (c) 982 (d) 990
31. 31.What is the units digit of 8^19? (a) 4 (b) 2 (c) 6 (d) 8
32. 32.What is the units digit of 4^42? (a) 6 (b) 4 (c) 7 (d) 5
33. 33.What is the smallest number you must add to 386 to make it divisible by 8? (a) 2 (b) 5 (c) 6 (d) 7
34. 34.What is the remainder when 324 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4
35. 35.What is the units digit of 2^37? (a) 2 (b) 4 (c) 8 (d) 6
36. 36.What is the remainder when 288 is divided by 7? (a) 2 (b) 1 (c) 0 (d) 6
37. 37.What is the smallest number you must add to 94 to make it divisible by 6? (a) 4 (b) 3 (c) 2 (d) 1
38. 38.What is the remainder when 227 is divided by 8? (a) 4 (b) 3 (c) 2 (d) 5
39. 39.What is the units digit of 7^46? (a) 7 (b) 1 (c) 3 (d) 9
40. 40.What is the smallest number you must add to 406 to make it divisible by 11? (a) 2 (b) 3 (c) 1 (d) 10

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