Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 233 · 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 7^13? (a) 1 (b) 9 (c) 7 (d) 3
2. 2.What is the remainder when 193 is divided by 3? (a) 0 (b) 1 (c) 2
3. 3.What is the remainder when 223 is divided by 3? (a) 2 (b) 0 (c) 1
4. 4.What is the smallest number you must add to 291 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
5. 5.Which of these numbers is divisible by 6? (a) 4938 (b) 4933 (c) 4949 (d) 4942
6. 6.Which of these numbers is divisible by 6? (a) 3350 (b) 3344 (c) 3338 (d) 3342
7. 7.Which of these numbers is divisible by 3? (a) 1765 (b) 1761 (c) 1763 (d) 1760
8. 8.What is the smallest number you must add to 438 to make it divisible by 10? (a) 8 (b) 3 (c) 1 (d) 2
9. 9.What is the remainder when 418 is divided by 9? (a) 5 (b) 3 (c) 6 (d) 4
10. 10.Which of these numbers is divisible by 4? (a) 1548 (b) 1553 (c) 1547 (d) 1550
11. 11.Which of these numbers is divisible by 8? (a) 809 (b) 794 (c) 800 (d) 801
12. 12.Which of these numbers is divisible by 11? (a) 8781 (b) 8760 (c) 8767 (d) 8770
13. 13.What is the smallest number you must add to 65 to make it divisible by 4? (a) 1 (b) 3 (c) 2
14. 14.What is the remainder when 580 is divided by 11? (a) 9 (b) 8 (c) 3 (d) 7
15. 15.Which of these numbers is divisible by 11? (a) 1502 (b) 1494 (c) 1485 (d) 1479
16. 16.What is the remainder when 230 is divided by 3? (a) 1 (b) 0 (c) 2
17. 17.What is the units digit of 8^50? (a) 2 (b) 8 (c) 4 (d) 6
18. 18.What is the remainder when 306 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
19. 19.What is the remainder when 34 is divided by 3? (a) 2 (b) 1 (c) 0
20. 20.What is the smallest number you must add to 789 to make it divisible by 10? (a) 2 (b) 9 (c) 3 (d) 1
21. 21.What is the remainder when 406 is divided by 9? (a) 2 (b) 1 (c) 8 (d) 0
22. 22.What is the units digit of 2^15? (a) 8 (b) 4 (c) 6 (d) 2
23. 23.What is the units digit of 7^10? (a) 7 (b) 3 (c) 1 (d) 9
24. 24.What is the units digit of 7^18? (a) 7 (b) 1 (c) 3 (d) 9
25. 25.Which of these numbers is divisible by 4? (a) 2853 (b) 2849 (c) 2845 (d) 2848
26. 26.What is the units digit of 2^16? (a) 6 (b) 8 (c) 2 (d) 4
27. 27.What is the smallest number you must add to 224 to make it divisible by 3? (a) 1 (b) 2
28. 28.What is the units digit of 9^56? (a) 9 (b) 1 (c) 2 (d) 0
29. 29.Which of these numbers is divisible by 3? (a) 2168 (b) 2174 (c) 2170 (d) 2169
30. 30.Which of these numbers is divisible by 6? (a) 3881 (b) 3870 (c) 3872 (d) 3865
31. 31.What is the smallest number you must add to 269 to make it divisible by 10? (a) 2 (b) 1 (c) 9 (d) 3
32. 32.What is the remainder when 87 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
33. 33.What is the smallest number you must add to 356 to make it divisible by 7? (a) 6 (b) 1 (c) 2 (d) 3
34. 34.What is the remainder when 251 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
35. 35.Which of these numbers is divisible by 11? (a) 6048 (b) 6050 (c) 6052 (d) 6070
36. 36.What is the remainder when 77 is divided by 3? (a) 0 (b) 2 (c) 1
37. 37.Which of these numbers is divisible by 8? (a) 1578 (b) 1584 (c) 1599 (d) 1588
38. 38.Which of these numbers is divisible by 4? (a) 2544 (b) 2551 (c) 2546 (d) 2542
39. 39.What is the smallest number you must add to 227 to make it divisible by 8? (a) 3 (b) 4 (c) 5 (d) 6
40. 40.What is the units digit of 4^23? (a) 3 (b) 4 (c) 6 (d) 5

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