Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 163 · 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 3^50? (a) 1 (b) 3 (c) 9 (d) 7
2. 2.What is the remainder when 368 is divided by 9? (a) 7 (b) 6 (c) 8 (d) 1
3. 3.Which of these numbers is divisible by 3? (a) 2273 (b) 2274 (c) 2279 (d) 2275
4. 4.What is the smallest number you must add to 110 to make it divisible by 7? (a) 2 (b) 3 (c) 1 (d) 5
5. 5.What is the units digit of 3^34? (a) 7 (b) 9 (c) 1 (d) 3
6. 6.Which of these numbers is divisible by 6? (a) 537 (b) 544 (c) 540 (d) 550
7. 7.Which of these numbers is divisible by 4? (a) 853 (b) 850 (c) 846 (d) 848
8. 8.Which of these numbers is divisible by 11? (a) 1394 (b) 1406 (c) 1409 (d) 1397
9. 9.Which of these numbers is divisible by 6? (a) 2899 (b) 2898 (c) 2906 (d) 2894
10. 10.What is the remainder when 340 is divided by 11? (a) 8 (b) 10 (c) 1 (d) 9
11. 11.Which of these numbers is divisible by 4? (a) 1543 (b) 1544 (c) 1550 (d) 1547
12. 12.What is the units digit of 3^19? (a) 1 (b) 7 (c) 3 (d) 9
13. 13.Which of these numbers is divisible by 9? (a) 3122 (b) 3114 (c) 3106 (d) 3131
14. 14.What is the smallest number you must add to 115 to make it divisible by 4? (a) 2 (b) 1 (c) 3
15. 15.What is the smallest number you must add to 142 to make it divisible by 9? (a) 2 (b) 1 (c) 7 (d) 3
16. 16.What is the smallest number you must add to 278 to make it divisible by 4? (a) 3 (b) 2 (c) 1
17. 17.What is the smallest number you must add to 215 to make it divisible by 7? (a) 2 (b) 5 (c) 1 (d) 3
18. 18.What is the units digit of 9^38? (a) 9 (b) 2 (c) 1 (d) 0
19. 19.What is the remainder when 282 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
20. 20.Which of these numbers is divisible by 8? (a) 3816 (b) 3809 (c) 3822 (d) 3826
21. 21.What is the smallest number you must add to 118 to make it divisible by 3? (a) 2 (b) 1
22. 22.What is the remainder when 183 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
23. 23.What is the units digit of 8^54? (a) 4 (b) 2 (c) 6 (d) 8
24. 24.What is the units digit of 4^38? (a) 4 (b) 7 (c) 5 (d) 6
25. 25.Which of these numbers is divisible by 4? (a) 3037 (b) 3042 (c) 3046 (d) 3040
26. 26.What is the units digit of 3^59? (a) 3 (b) 9 (c) 1 (d) 7
27. 27.What is the remainder when 285 is divided by 7? (a) 5 (b) 6 (c) 4 (d) 2
28. 28.What is the remainder when 193 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
29. 29.Which of these numbers is divisible by 9? (a) 3731 (b) 3710 (c) 3718 (d) 3717
30. 30.What is the remainder when 461 is divided by 12? (a) 7 (b) 6 (c) 4 (d) 5
31. 31.Which of these numbers is divisible by 9? (a) 1810 (b) 1806 (c) 1820 (d) 1809
32. 32.Which of these numbers is divisible by 9? (a) 5089 (b) 5078 (c) 5085 (d) 5098
33. 33.What is the smallest number you must add to 176 to make it divisible by 3? (a) 1 (b) 2
34. 34.Which of these numbers is divisible by 8? (a) 4642 (b) 4640 (c) 4637 (d) 4649
35. 35.What is the units digit of 2^48? (a) 8 (b) 6 (c) 4 (d) 2
36. 36.What is the units digit of 4^28? (a) 4 (b) 5 (c) 6 (d) 7
37. 37.Which of these numbers is divisible by 8? (a) 3932 (b) 3927 (c) 3928 (d) 3943
38. 38.What is the units digit of 3^12? (a) 7 (b) 3 (c) 1 (d) 9
39. 39.Which of these numbers is divisible by 3? (a) 1110 (b) 1114 (c) 1109 (d) 1111
40. 40.What is the units digit of 9^30? (a) 9 (b) 2 (c) 1 (d) 0

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