Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 207 · 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 37 is divided by 3? (a) 1 (b) 2 (c) 0
2. 2.What is the smallest number you must add to 442 to make it divisible by 10? (a) 7 (b) 2 (c) 9 (d) 8
3. 3.Which of these numbers is divisible by 8? (a) 840 (b) 847 (c) 837 (d) 855
4. 4.What is the smallest number you must add to 61 to make it divisible by 3? (a) 2 (b) 1
5. 5.What is the remainder when 109 is divided by 6? (a) 0 (b) 5 (c) 2 (d) 1
6. 6.What is the units digit of 3^44? (a) 3 (b) 9 (c) 7 (d) 1
7. 7.What is the remainder when 181 is divided by 7? (a) 5 (b) 1 (c) 4 (d) 6
8. 8.What is the units digit of 9^26? (a) 1 (b) 2 (c) 0 (d) 9
9. 9.What is the smallest number you must add to 388 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
10. 10.What is the units digit of 8^20? (a) 6 (b) 8 (c) 2 (d) 4
11. 11.What is the remainder when 567 is divided by 12? (a) 9 (b) 2 (c) 3 (d) 4
12. 12.Which of these numbers is divisible by 9? (a) 4114 (b) 4099 (c) 4110 (d) 4104
13. 13.Which of these numbers is divisible by 9? (a) 2502 (b) 2494 (c) 2510 (d) 2515
14. 14.Which of these numbers is divisible by 8? (a) 3667 (b) 3672 (c) 3677 (d) 3687
15. 15.What is the smallest number you must add to 47 to make it divisible by 3? (a) 1 (b) 2
16. 16.Which of these numbers is divisible by 3? (a) 1061 (b) 1058 (c) 1056 (d) 1055
17. 17.What is the units digit of 3^55? (a) 1 (b) 3 (c) 9 (d) 7
18. 18.What is the remainder when 278 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
19. 19.What is the units digit of 2^46? (a) 4 (b) 8 (c) 6 (d) 2
20. 20.Which of these numbers is divisible by 6? (a) 5123 (b) 5107 (c) 5114 (d) 5112
21. 21.What is the units digit of 8^50? (a) 2 (b) 6 (c) 8 (d) 4
22. 22.What is the units digit of 4^11? (a) 5 (b) 6 (c) 3 (d) 4
23. 23.Which of these numbers is divisible by 11? (a) 6292 (b) 6300 (c) 6310 (d) 6287
24. 24.What is the units digit of 3^30? (a) 3 (b) 1 (c) 7 (d) 9
25. 25.What is the remainder when 299 is divided by 9? (a) 3 (b) 7 (c) 1 (d) 2
26. 26.Which of these numbers is divisible by 3? (a) 2296 (b) 2302 (c) 2300 (d) 2298
27. 27.Which of these numbers is divisible by 3? (a) 2035 (b) 2031 (c) 2029 (d) 2033
28. 28.What is the smallest number you must add to 267 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
29. 29.What is the units digit of 8^27? (a) 4 (b) 8 (c) 6 (d) 2
30. 30.What is the remainder when 272 is divided by 7? (a) 4 (b) 5 (c) 1 (d) 6
31. 31.Which of these numbers is divisible by 4? (a) 2761 (b) 2767 (c) 2769 (d) 2764
32. 32.Which of these numbers is divisible by 3? (a) 2633 (b) 2626 (c) 2629 (d) 2628
33. 33.What is the remainder when 734 is divided by 10? (a) 5 (b) 3 (c) 4 (d) 6
34. 34.What is the remainder when 121 is divided by 3? (a) 0 (b) 1 (c) 2
35. 35.Which of these numbers is divisible by 6? (a) 1548 (b) 1550 (c) 1555 (d) 1546
36. 36.What is the smallest number you must add to 170 to make it divisible by 9? (a) 2 (b) 3 (c) 1 (d) 8
37. 37.What is the units digit of 8^10? (a) 2 (b) 8 (c) 4 (d) 6
38. 38.What is the units digit of 2^26? (a) 2 (b) 4 (c) 6 (d) 8
39. 39.What is the remainder when 343 is divided by 8? (a) 5 (b) 6 (c) 7 (d) 1
40. 40.What is the units digit of 2^17? (a) 6 (b) 8 (c) 2 (d) 4

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