Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 110 · 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 395 is divided by 10? (a) 4 (b) 5 (c) 6 (d) 7
2. 2.What is the remainder when 158 is divided by 7? (a) 3 (b) 6 (c) 5 (d) 4
3. 3.What is the units digit of 3^26? (a) 3 (b) 9 (c) 7 (d) 1
4. 4.Which of these numbers is divisible by 4? (a) 308 (b) 313 (c) 306 (d) 311
5. 5.Which of these numbers is divisible by 8? (a) 6009 (b) 6000 (c) 6004 (d) 5993
6. 6.What is the smallest number you must add to 383 to make it divisible by 10? (a) 7 (b) 8 (c) 3 (d) 6
7. 7.What is the remainder when 176 is divided by 10? (a) 7 (b) 5 (c) 6 (d) 4
8. 8.What is the smallest number you must add to 35 to make it divisible by 3? (a) 1 (b) 2
9. 9.What is the remainder when 114 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
10. 10.What is the remainder when 71 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
11. 11.What is the units digit of 7^60? (a) 3 (b) 7 (c) 1 (d) 9
12. 12.What is the smallest number you must add to 527 to make it divisible by 12? (a) 11 (b) 1 (c) 2 (d) 3
13. 13.What is the remainder when 688 is divided by 9? (a) 4 (b) 6 (c) 5 (d) 3
14. 14.Which of these numbers is divisible by 3? (a) 1072 (b) 1075 (c) 1071 (d) 1069
15. 15.What is the smallest number you must add to 138 to make it divisible by 12? (a) 7 (b) 5 (c) 8 (d) 6
16. 16.What is the remainder when 163 is divided by 3? (a) 0 (b) 2 (c) 1
17. 17.What is the remainder when 146 is divided by 3? (a) 2 (b) 0 (c) 1
18. 18.What is the units digit of 7^48? (a) 9 (b) 1 (c) 3 (d) 7
19. 19.What is the units digit of 2^29? (a) 2 (b) 4 (c) 8 (d) 6
20. 20.What is the remainder when 130 is divided by 11? (a) 8 (b) 9 (c) 2 (d) 10
21. 21.Which of these numbers is divisible by 4? (a) 2273 (b) 2278 (c) 2283 (d) 2276
22. 22.What is the units digit of 2^22? (a) 4 (b) 2 (c) 8 (d) 6
23. 23.Which of these numbers is divisible by 6? (a) 4978 (b) 4982 (c) 4971 (d) 4974
24. 24.What is the remainder when 270 is divided by 7? (a) 4 (b) 6 (c) 3 (d) 5
25. 25.What is the units digit of 3^15? (a) 1 (b) 9 (c) 3 (d) 7
26. 26.What is the remainder when 261 is divided by 10? (a) 9 (b) 1 (c) 0 (d) 2
27. 27.What is the remainder when 122 is divided by 3? (a) 2 (b) 0 (c) 1
28. 28.What is the units digit of 3^52? (a) 9 (b) 7 (c) 1 (d) 3
29. 29.What is the smallest number you must add to 856 to make it divisible by 12? (a) 8 (b) 9 (c) 4 (d) 7
30. 30.Which of these numbers is divisible by 6? (a) 1195 (b) 1189 (c) 1183 (d) 1188
31. 31.Which of these numbers is divisible by 8? (a) 5174 (b) 5183 (c) 5164 (d) 5168
32. 32.What is the remainder when 562 is divided by 10? (a) 2 (b) 8 (c) 1 (d) 3
33. 33.What is the remainder when 77 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
34. 34.What is the units digit of 9^11? (a) 7 (b) 1 (c) 9 (d) 8
35. 35.What is the units digit of 7^54? (a) 1 (b) 3 (c) 7 (d) 9
36. 36.What is the units digit of 9^49? (a) 9 (b) 1 (c) 8 (d) 7
37. 37.Which of these numbers is divisible by 6? (a) 1683 (b) 1680 (c) 1687 (d) 1676
38. 38.Which of these numbers is divisible by 8? (a) 2208 (b) 2203 (c) 2211 (d) 2223
39. 39.What is the units digit of 9^34? (a) 2 (b) 0 (c) 1 (d) 9
40. 40.What is the smallest number you must add to 146 to make it divisible by 10? (a) 6 (b) 3 (c) 4 (d) 5

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