Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 31 · 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 4^17? (a) 5 (b) 3 (c) 6 (d) 4
2. 2.Which of these numbers is divisible by 9? (a) 891 (b) 901 (c) 895 (d) 889
3. 3.Which of these numbers is divisible by 4? (a) 1269 (b) 1264 (c) 1266 (d) 1263
4. 4.Which of these numbers is divisible by 11? (a) 4466 (b) 4481 (c) 4464 (d) 4475
5. 5.What is the units digit of 7^10? (a) 7 (b) 1 (c) 3 (d) 9
6. 6.What is the smallest number you must add to 124 to make it divisible by 7? (a) 2 (b) 1 (c) 3 (d) 5
7. 7.Which of these numbers is divisible by 6? (a) 1291 (b) 1287 (c) 1290 (d) 1301
8. 8.What is the smallest number you must add to 690 to make it divisible by 11? (a) 2 (b) 3 (c) 4 (d) 8
9. 9.Which of these numbers is divisible by 4? (a) 3041 (b) 3039 (c) 3040 (d) 3046
10. 10.What is the units digit of 4^51? (a) 3 (b) 4 (c) 5 (d) 6
11. 11.What is the smallest number you must add to 139 to make it divisible by 3? (a) 1 (b) 2
12. 12.What is the remainder when 95 is divided by 6? (a) 4 (b) 1 (c) 5 (d) 3
13. 13.What is the remainder when 533 is divided by 7? (a) 0 (b) 1 (c) 2 (d) 6
14. 14.What is the units digit of 7^11? (a) 7 (b) 9 (c) 3 (d) 1
15. 15.What is the smallest number you must add to 934 to make it divisible by 12? (a) 3 (b) 2 (c) 1 (d) 10
16. 16.What is the smallest number you must add to 260 to make it divisible by 8? (a) 4 (b) 3 (c) 6 (d) 5
17. 17.Which of these numbers is divisible by 11? (a) 5342 (b) 5335 (c) 5332 (d) 5352
18. 18.What is the units digit of 8^37? (a) 8 (b) 2 (c) 4 (d) 6
19. 19.What is the smallest number you must add to 636 to make it divisible by 11? (a) 2 (b) 3 (c) 9 (d) 1
20. 20.Which of these numbers is divisible by 8? (a) 5960 (b) 5969 (c) 5961 (d) 5957
21. 21.Which of these numbers is divisible by 4? (a) 245 (b) 243 (c) 249 (d) 244
22. 22.Which of these numbers is divisible by 11? (a) 9867 (b) 9888 (c) 9860 (d) 9871
23. 23.What is the units digit of 4^19? (a) 5 (b) 6 (c) 4 (d) 3
24. 24.What is the units digit of 2^44? (a) 6 (b) 2 (c) 4 (d) 8
25. 25.Which of these numbers is divisible by 3? (a) 698 (b) 694 (c) 696 (d) 701
26. 26.What is the units digit of 8^40? (a) 6 (b) 2 (c) 4 (d) 8
27. 27.What is the smallest number you must add to 173 to make it divisible by 4? (a) 2 (b) 1 (c) 3
28. 28.What is the smallest number you must add to 215 to make it divisible by 6? (a) 3 (b) 1 (c) 2 (d) 5
29. 29.Which of these numbers is divisible by 3? (a) 570 (b) 574 (c) 568 (d) 571
30. 30.Which of these numbers is divisible by 11? (a) 5847 (b) 5829 (c) 5836 (d) 5830
31. 31.What is the smallest number you must add to 161 to make it divisible by 3? (a) 2 (b) 1
32. 32.Which of these numbers is divisible by 11? (a) 2606 (b) 2589 (c) 2596 (d) 2614
33. 33.Which of these numbers is divisible by 3? (a) 2051 (b) 2053 (c) 2047 (d) 2049
34. 34.What is the smallest number you must add to 213 to make it divisible by 4? (a) 1 (b) 3 (c) 2
35. 35.Which of these numbers is divisible by 9? (a) 6448 (b) 6438 (c) 6456 (d) 6444
36. 36.What is the units digit of 4^50? (a) 4 (b) 7 (c) 5 (d) 6
37. 37.Which of these numbers is divisible by 4? (a) 2629 (b) 2627 (c) 2624 (d) 2622
38. 38.What is the smallest number you must add to 289 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
39. 39.What is the smallest number you must add to 151 to make it divisible by 9? (a) 1 (b) 2 (c) 7 (d) 3
40. 40.What is the units digit of 9^16? (a) 0 (b) 9 (c) 1 (d) 2

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