Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 848 · 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 4^21? (a) 6 (b) 3 (c) 5 (d) 4
2. 2.What is the units digit of 9^13? (a) 8 (b) 9 (c) 1 (d) 7
3. 3.What is the remainder when 236 is divided by 3? (a) 0 (b) 2 (c) 1
4. 4.What is the units digit of 7^46? (a) 1 (b) 3 (c) 7 (d) 9
5. 5.Which of these numbers is divisible by 6? (a) 289 (b) 288 (c) 284 (d) 296
6. 6.What is the smallest number you must add to 64 to make it divisible by 3? (a) 1 (b) 2
7. 7.What is the remainder when 57 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
8. 8.Which of these numbers is divisible by 4? (a) 206 (b) 200 (c) 202 (d) 198
9. 9.What is the smallest number you must add to 242 to make it divisible by 4? (a) 1 (b) 3 (c) 2
10. 10.What is the units digit of 4^10? (a) 5 (b) 7 (c) 6 (d) 4
11. 11.What is the smallest number you must add to 198 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
12. 12.What is the units digit of 3^11? (a) 3 (b) 9 (c) 7 (d) 1
13. 13.What is the units digit of 3^25? (a) 3 (b) 1 (c) 7 (d) 9
14. 14.What is the remainder when 351 is divided by 5? (a) 0 (b) 2 (c) 4 (d) 1
15. 15.What is the units digit of 7^39? (a) 1 (b) 3 (c) 9 (d) 7
16. 16.What is the smallest number you must add to 174 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
17. 17.What is the units digit of 3^14? (a) 9 (b) 3 (c) 7 (d) 1
18. 18.Which of these numbers is divisible by 6? (a) 2118 (b) 2115 (c) 2127 (d) 2122
19. 19.What is the units digit of 8^56? (a) 2 (b) 6 (c) 8 (d) 4
20. 20.What is the remainder when 69 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
21. 21.Which of these numbers is divisible by 3? (a) 2330 (b) 2325 (c) 2326 (d) 2323
22. 22.Which of these numbers is divisible by 9? (a) 930 (b) 918 (c) 910 (d) 919
23. 23.What is the smallest number you must add to 118 to make it divisible by 4? (a) 3 (b) 2 (c) 1
24. 24.Which of these numbers is divisible by 3? (a) 881 (b) 877 (c) 884 (d) 879
25. 25.Which of these numbers is divisible by 9? (a) 2466 (b) 2474 (c) 2483 (d) 2465
26. 26.What is the units digit of 2^59? (a) 2 (b) 6 (c) 4 (d) 8
27. 27.What is the smallest number you must add to 221 to make it divisible by 3? (a) 2 (b) 1
28. 28.What is the units digit of 7^33? (a) 9 (b) 7 (c) 1 (d) 3
29. 29.What is the units digit of 3^38? (a) 7 (b) 9 (c) 1 (d) 3
30. 30.What is the smallest number you must add to 718 to make it divisible by 10? (a) 8 (b) 3 (c) 2 (d) 1
31. 31.Which of these numbers is divisible by 9? (a) 734 (b) 722 (c) 720 (d) 717
32. 32.What is the remainder when 918 is divided by 12? (a) 6 (b) 7 (c) 5 (d) 8
33. 33.Which of these numbers is divisible by 8? (a) 4059 (b) 4056 (c) 4054 (d) 4070
34. 34.What is the units digit of 3^23? (a) 3 (b) 7 (c) 1 (d) 9
35. 35.What is the units digit of 4^40? (a) 4 (b) 7 (c) 6 (d) 5
36. 36.What is the remainder when 760 is divided by 11? (a) 10 (b) 2 (c) 1 (d) 0
37. 37.Which of these numbers is divisible by 4? (a) 1852 (b) 1855 (c) 1851 (d) 1857
38. 38.What is the remainder when 314 is divided by 10? (a) 6 (b) 3 (c) 4 (d) 5
39. 39.Which of these numbers is divisible by 8? (a) 6857 (b) 6856 (c) 6855 (d) 6866
40. 40.What is the remainder when 283 is divided by 10? (a) 4 (b) 3 (c) 7 (d) 2

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