Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 564 · 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 4^35? (a) 3 (b) 5 (c) 4 (d) 6
2. 2.What is the smallest number you must add to 242 to make it divisible by 4? (a) 2 (b) 1 (c) 3
3. 3.Which of these numbers is divisible by 9? (a) 1637 (b) 1638 (c) 1644 (d) 1648
4. 4.What is the smallest number you must add to 766 to make it divisible by 10? (a) 4 (b) 5 (c) 6 (d) 3
5. 5.What is the units digit of 3^25? (a) 1 (b) 7 (c) 9 (d) 3
6. 6.What is the smallest number you must add to 53 to make it divisible by 3? (a) 2 (b) 1
7. 7.What is the remainder when 227 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
8. 8.Which of these numbers is divisible by 6? (a) 3739 (b) 3735 (c) 3727 (d) 3732
9. 9.What is the units digit of 2^35? (a) 2 (b) 8 (c) 6 (d) 4
10. 10.What is the smallest number you must add to 401 to make it divisible by 9? (a) 4 (b) 3 (c) 6 (d) 5
11. 11.What is the smallest number you must add to 289 to make it divisible by 6? (a) 1 (b) 4 (c) 5 (d) 3
12. 12.Which of these numbers is divisible by 3? (a) 1034 (b) 1037 (c) 1039 (d) 1035
13. 13.What is the smallest number you must add to 171 to make it divisible by 7? (a) 5 (b) 4 (c) 3 (d) 6
14. 14.Which of these numbers is divisible by 4? (a) 2026 (b) 2019 (c) 2020 (d) 2022
15. 15.What is the units digit of 2^50? (a) 8 (b) 2 (c) 4 (d) 6
16. 16.What is the remainder when 281 is divided by 9? (a) 1 (b) 7 (c) 2 (d) 3
17. 17.What is the smallest number you must add to 285 to make it divisible by 12? (a) 4 (b) 9 (c) 3 (d) 2
18. 18.Which of these numbers is divisible by 9? (a) 3347 (b) 3353 (c) 3362 (d) 3348
19. 19.What is the remainder when 98 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
20. 20.What is the smallest number you must add to 41 to make it divisible by 3? (a) 1 (b) 2
21. 21.What is the remainder when 674 is divided by 10? (a) 6 (b) 4 (c) 3 (d) 5
22. 22.What is the units digit of 2^56? (a) 8 (b) 4 (c) 2 (d) 6
23. 23.What is the remainder when 807 is divided by 10? (a) 6 (b) 7 (c) 8 (d) 3
24. 24.What is the remainder when 161 is divided by 3? (a) 1 (b) 2 (c) 0
25. 25.Which of these numbers is divisible by 4? (a) 1329 (b) 1323 (c) 1326 (d) 1324
26. 26.What is the remainder when 229 is divided by 6? (a) 1 (b) 5 (c) 2 (d) 0
27. 27.Which of these numbers is divisible by 11? (a) 4192 (b) 4189 (c) 4211 (d) 4191
28. 28.Which of these numbers is divisible by 4? (a) 1670 (b) 1674 (c) 1678 (d) 1672
29. 29.What is the units digit of 2^12? (a) 6 (b) 2 (c) 4 (d) 8
30. 30.What is the smallest number you must add to 292 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
31. 31.What is the remainder when 61 is divided by 3? (a) 1 (b) 0 (c) 2
32. 32.What is the units digit of 3^53? (a) 3 (b) 7 (c) 1 (d) 9
33. 33.Which of these numbers is divisible by 4? (a) 847 (b) 853 (c) 850 (d) 848
34. 34.Which of these numbers is divisible by 4? (a) 2635 (b) 2637 (c) 2636 (d) 2642
35. 35.What is the remainder when 69 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
36. 36.What is the remainder when 209 is divided by 10? (a) 9 (b) 1 (c) 7 (d) 8
37. 37.Which of these numbers is divisible by 4? (a) 538 (b) 533 (c) 536 (d) 543
38. 38.Which of these numbers is divisible by 9? (a) 1970 (b) 1962 (c) 1978 (d) 1958
39. 39.What is the units digit of 3^55? (a) 3 (b) 7 (c) 9 (d) 1
40. 40.What is the units digit of 3^57? (a) 7 (b) 3 (c) 9 (d) 1

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