Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 402 · 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 160 is divided by 12? (a) 3 (b) 5 (c) 8 (d) 4
2. 2.Which of these numbers is divisible by 4? (a) 3031 (b) 3021 (c) 3024 (d) 3025
3. 3.What is the units digit of 7^32? (a) 3 (b) 1 (c) 7 (d) 9
4. 4.What is the smallest number you must add to 578 to make it divisible by 9? (a) 2 (b) 7 (c) 8 (d) 6
5. 5.What is the remainder when 414 is divided by 7? (a) 1 (b) 2 (c) 0 (d) 6
6. 6.What is the remainder when 134 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
7. 7.What is the units digit of 7^14? (a) 1 (b) 7 (c) 9 (d) 3
8. 8.What is the remainder when 592 is divided by 9? (a) 8 (b) 2 (c) 7 (d) 6
9. 9.Which of these numbers is divisible by 6? (a) 2069 (b) 2064 (c) 2059 (d) 2074
10. 10.What is the smallest number you must add to 358 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
11. 11.What is the remainder when 768 is divided by 11? (a) 2 (b) 9 (c) 8 (d) 10
12. 12.What is the units digit of 7^22? (a) 7 (b) 3 (c) 1 (d) 9
13. 13.Which of these numbers is divisible by 3? (a) 1323 (b) 1321 (c) 1325 (d) 1328
14. 14.Which of these numbers is divisible by 11? (a) 9355 (b) 9342 (c) 9362 (d) 9350
15. 15.What is the units digit of 4^49? (a) 5 (b) 6 (c) 4 (d) 3
16. 16.What is the remainder when 144 is divided by 5? (a) 3 (b) 4 (c) 2 (d) 1
17. 17.Which of these numbers is divisible by 11? (a) 1827 (b) 1825 (c) 1807 (d) 1815
18. 18.Which of these numbers is divisible by 9? (a) 4734 (b) 4745 (c) 4733 (d) 4736
19. 19.What is the units digit of 7^31? (a) 1 (b) 9 (c) 7 (d) 3
20. 20.What is the remainder when 479 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 5
21. 21.What is the remainder when 247 is divided by 7? (a) 2 (b) 3 (c) 1 (d) 5
22. 22.Which of these numbers is divisible by 3? (a) 2138 (b) 2141 (c) 2134 (d) 2136
23. 23.Which of these numbers is divisible by 3? (a) 2499 (b) 2500 (c) 2504 (d) 2497
24. 24.What is the units digit of 8^51? (a) 6 (b) 8 (c) 2 (d) 4
25. 25.What is the remainder when 463 is divided by 9? (a) 5 (b) 3 (c) 6 (d) 4
26. 26.What is the smallest number you must add to 644 to make it divisible by 8? (a) 5 (b) 4 (c) 3 (d) 6
27. 27.What is the remainder when 182 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
28. 28.Which of these numbers is divisible by 4? (a) 187 (b) 180 (c) 181 (d) 178
29. 29.Which of these numbers is divisible by 6? (a) 5361 (b) 5358 (c) 5367 (d) 5353
30. 30.Which of these numbers is divisible by 11? (a) 6664 (b) 6680 (c) 6674 (d) 6666
31. 31.What is the units digit of 9^24? (a) 0 (b) 9 (c) 2 (d) 1
32. 32.What is the remainder when 145 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
33. 33.What is the remainder when 478 is divided by 10? (a) 8 (b) 9 (c) 7 (d) 2
34. 34.Which of these numbers is divisible by 8? (a) 2890 (b) 2905 (c) 2896 (d) 2898
35. 35.Which of these numbers is divisible by 8? (a) 5275 (b) 5268 (c) 5285 (d) 5272
36. 36.What is the smallest number you must add to 579 to make it divisible by 11? (a) 4 (b) 5 (c) 7 (d) 3
37. 37.What is the units digit of 4^56? (a) 4 (b) 5 (c) 6 (d) 7
38. 38.Which of these numbers is divisible by 3? (a) 2681 (b) 2679 (c) 2683 (d) 2678
39. 39.Which of these numbers is divisible by 6? (a) 3313 (b) 3308 (c) 3312 (d) 3321
40. 40.Which of these numbers is divisible by 3? (a) 1220 (b) 1217 (c) 1213 (d) 1215

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