Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 4 · 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 2^25? (a) 6 (b) 8 (c) 2 (d) 4
2. 2.What is the units digit of 8^22? (a) 6 (b) 4 (c) 2 (d) 8
3. 3.What is the units digit of 4^56? (a) 7 (b) 5 (c) 6 (d) 4
4. 4.Which of these numbers is divisible by 3? (a) 460 (b) 464 (c) 459 (d) 458
5. 5.What is the remainder when 461 is divided by 6? (a) 4 (b) 3 (c) 5 (d) 1
6. 6.What is the smallest number you must add to 109 to make it divisible by 3? (a) 2 (b) 1
7. 7.What is the remainder when 254 is divided by 7? (a) 1 (b) 2 (c) 3 (d) 5
8. 8.What is the remainder when 163 is divided by 3? (a) 2 (b) 0 (c) 1
9. 9.What is the smallest number you must add to 587 to make it divisible by 10? (a) 2 (b) 4 (c) 3 (d) 7
10. 10.What is the units digit of 3^20? (a) 9 (b) 7 (c) 3 (d) 1
11. 11.What is the smallest number you must add to 443 to make it divisible by 10? (a) 3 (b) 8 (c) 7 (d) 6
12. 12.What is the units digit of 4^38? (a) 4 (b) 5 (c) 7 (d) 6
13. 13.What is the smallest number you must add to 398 to make it divisible by 10? (a) 3 (b) 8 (c) 2 (d) 1
14. 14.Which of these numbers is divisible by 8? (a) 1211 (b) 1201 (c) 1208 (d) 1221
15. 15.Which of these numbers is divisible by 6? (a) 4842 (b) 4844 (c) 4850 (d) 4837
16. 16.What is the smallest number you must add to 251 to make it divisible by 9? (a) 2 (b) 8 (c) 1 (d) 3
17. 17.What is the remainder when 867 is divided by 12? (a) 9 (b) 3 (c) 2 (d) 4
18. 18.What is the remainder when 791 is divided by 10? (a) 9 (b) 2 (c) 0 (d) 1
19. 19.What is the remainder when 34 is divided by 3? (a) 0 (b) 2 (c) 1
20. 20.Which of these numbers is divisible by 8? (a) 6860 (b) 6842 (c) 6848 (d) 6851
21. 21.Which of these numbers is divisible by 4? (a) 270 (b) 266 (c) 263 (d) 264
22. 22.Which of these numbers is divisible by 3? (a) 200 (b) 193 (c) 195 (d) 196
23. 23.What is the units digit of 2^33? (a) 6 (b) 8 (c) 4 (d) 2
24. 24.What is the smallest number you must add to 90 to make it divisible by 7? (a) 3 (b) 6 (c) 1 (d) 2
25. 25.What is the smallest number you must add to 142 to make it divisible by 4? (a) 1 (b) 2 (c) 3
26. 26.What is the units digit of 4^46? (a) 4 (b) 6 (c) 5 (d) 7
27. 27.Which of these numbers is divisible by 3? (a) 2023 (b) 2025 (c) 2026 (d) 2030
28. 28.What is the units digit of 3^38? (a) 1 (b) 7 (c) 9 (d) 3
29. 29.Which of these numbers is divisible by 3? (a) 631 (b) 634 (c) 638 (d) 633
30. 30.What is the smallest number you must add to 372 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
31. 31.What is the smallest number you must add to 254 to make it divisible by 10? (a) 4 (b) 6 (c) 7 (d) 5
32. 32.What is the units digit of 4^51? (a) 4 (b) 5 (c) 3 (d) 6
33. 33.What is the units digit of 8^15? (a) 4 (b) 6 (c) 8 (d) 2
34. 34.Which of these numbers is divisible by 8? (a) 3324 (b) 3320 (c) 3329 (d) 3314
35. 35.What is the smallest number you must add to 62 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
36. 36.Which of these numbers is divisible by 11? (a) 1841 (b) 1823 (c) 1826 (d) 1832
37. 37.Which of these numbers is divisible by 3? (a) 1225 (b) 1228 (c) 1227 (d) 1231
38. 38.Which of these numbers is divisible by 9? (a) 6630 (b) 6618 (c) 6636 (d) 6624
39. 39.Which of these numbers is divisible by 3? (a) 592 (b) 596 (c) 591 (d) 590
40. 40.Which of these numbers is divisible by 3? (a) 2668 (b) 2666 (c) 2662 (d) 2664

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