Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 747 · 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 smallest number you must add to 208 to make it divisible by 11? (a) 1 (b) 10 (c) 2 (d) 3
2. 2.Which of these numbers is divisible by 9? (a) 2800 (b) 2816 (c) 2799 (d) 2796
3. 3.What is the units digit of 8^26? (a) 2 (b) 8 (c) 4 (d) 6
4. 4.What is the remainder when 334 is divided by 9? (a) 8 (b) 1 (c) 2 (d) 0
5. 5.What is the units digit of 2^17? (a) 2 (b) 6 (c) 4 (d) 8
6. 6.What is the smallest number you must add to 681 to make it divisible by 12? (a) 9 (b) 4 (c) 2 (d) 3
7. 7.What is the units digit of 2^46? (a) 4 (b) 6 (c) 2 (d) 8
8. 8.Which of these numbers is divisible by 6? (a) 289 (b) 287 (c) 296 (d) 288
9. 9.What is the smallest number you must add to 242 to make it divisible by 4? (a) 2 (b) 3 (c) 1
10. 10.What is the smallest number you must add to 85 to make it divisible by 3? (a) 1 (b) 2
11. 11.What is the smallest number you must add to 124 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
12. 12.What is the units digit of 3^49? (a) 1 (b) 9 (c) 3 (d) 7
13. 13.What is the units digit of 4^56? (a) 7 (b) 4 (c) 5 (d) 6
14. 14.Which of these numbers is divisible by 9? (a) 4977 (b) 4994 (c) 4976 (d) 4980
15. 15.Which of these numbers is divisible by 11? (a) 2633 (b) 2642 (c) 2652 (d) 2640
16. 16.What is the smallest number you must add to 157 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
17. 17.What is the remainder when 543 is divided by 9? (a) 6 (b) 4 (c) 2 (d) 3
18. 18.What is the remainder when 107 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
19. 19.Which of these numbers is divisible by 8? (a) 2889 (b) 2903 (c) 2896 (d) 2910
20. 20.Which of these numbers is divisible by 4? (a) 981 (b) 975 (c) 976 (d) 979
21. 21.Which of these numbers is divisible by 8? (a) 3891 (b) 3898 (c) 3888 (d) 3886
22. 22.Which of these numbers is divisible by 4? (a) 1193 (b) 1188 (c) 1191 (d) 1187
23. 23.What is the smallest number you must add to 328 to make it divisible by 10? (a) 3 (b) 8 (c) 1 (d) 2
24. 24.What is the units digit of 7^53? (a) 9 (b) 7 (c) 3 (d) 1
25. 25.What is the units digit of 7^43? (a) 7 (b) 3 (c) 1 (d) 9
26. 26.What is the smallest number you must add to 311 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
27. 27.What is the units digit of 7^58? (a) 7 (b) 9 (c) 1 (d) 3
28. 28.Which of these numbers is divisible by 3? (a) 229 (b) 228 (c) 227 (d) 232
29. 29.What is the units digit of 3^13? (a) 3 (b) 7 (c) 9 (d) 1
30. 30.What is the remainder when 467 is divided by 11? (a) 6 (b) 4 (c) 5 (d) 7
31. 31.What is the remainder when 696 is divided by 11? (a) 8 (b) 4 (c) 3 (d) 2
32. 32.What is the remainder when 317 is divided by 6? (a) 3 (b) 4 (c) 1 (d) 5
33. 33.Which of these numbers is divisible by 11? (a) 7568 (b) 7558 (c) 7587 (d) 7571
34. 34.Which of these numbers is divisible by 6? (a) 3876 (b) 3874 (c) 3880 (d) 3885
35. 35.Which of these numbers is divisible by 9? (a) 6615 (b) 6621 (c) 6614 (d) 6632
36. 36.What is the units digit of 2^13? (a) 8 (b) 6 (c) 4 (d) 2
37. 37.What is the remainder when 427 is divided by 11? (a) 8 (b) 10 (c) 9 (d) 2
38. 38.Which of these numbers is divisible by 9? (a) 6236 (b) 6213 (c) 6224 (d) 6219
39. 39.Which of these numbers is divisible by 6? (a) 1311 (b) 1325 (c) 1319 (d) 1314
40. 40.What is the smallest number you must add to 173 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1

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