Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 694 · 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.Which of these numbers is divisible by 11? (a) 8374 (b) 8366 (c) 8391 (d) 8371
2. 2.Which of these numbers is divisible by 9? (a) 6110 (b) 6099 (c) 6102 (d) 6114
3. 3.Which of these numbers is divisible by 11? (a) 3543 (b) 3539 (c) 3562 (d) 3542
4. 4.Which of these numbers is divisible by 3? (a) 146 (b) 144 (c) 143 (d) 149
5. 5.What is the units digit of 7^44? (a) 7 (b) 9 (c) 1 (d) 3
6. 6.What is the units digit of 4^30? (a) 7 (b) 6 (c) 4 (d) 5
7. 7.Which of these numbers is divisible by 6? (a) 325 (b) 322 (c) 332 (d) 324
8. 8.What is the units digit of 2^10? (a) 8 (b) 6 (c) 4 (d) 2
9. 9.What is the units digit of 9^33? (a) 9 (b) 1 (c) 8 (d) 7
10. 10.Which of these numbers is divisible by 6? (a) 1090 (b) 1096 (c) 1086 (d) 1083
11. 11.What is the remainder when 377 is divided by 11? (a) 4 (b) 2 (c) 8 (d) 3
12. 12.What is the units digit of 4^25? (a) 3 (b) 5 (c) 6 (d) 4
13. 13.Which of these numbers is divisible by 11? (a) 5291 (b) 5305 (c) 5299 (d) 5284
14. 14.What is the units digit of 3^14? (a) 1 (b) 9 (c) 3 (d) 7
15. 15.What is the units digit of 3^45? (a) 7 (b) 3 (c) 9 (d) 1
16. 16.What is the smallest number you must add to 578 to make it divisible by 8? (a) 6 (b) 5 (c) 2 (d) 7
17. 17.What is the smallest number you must add to 387 to make it divisible by 8? (a) 5 (b) 3 (c) 4 (d) 6
18. 18.What is the units digit of 8^19? (a) 4 (b) 8 (c) 2 (d) 6
19. 19.Which of these numbers is divisible by 8? (a) 582 (b) 563 (c) 568 (d) 573
20. 20.What is the remainder when 167 is divided by 11? (a) 9 (b) 1 (c) 3 (d) 2
21. 21.Which of these numbers is divisible by 3? (a) 970 (b) 967 (c) 969 (d) 973
22. 22.What is the remainder when 58 is divided by 3? (a) 2 (b) 0 (c) 1
23. 23.What is the units digit of 9^32? (a) 9 (b) 0 (c) 2 (d) 1
24. 24.Which of these numbers is divisible by 8? (a) 981 (b) 993 (c) 986 (d) 984
25. 25.What is the remainder when 56 is divided by 3? (a) 0 (b) 2 (c) 1
26. 26.What is the units digit of 7^51? (a) 7 (b) 1 (c) 9 (d) 3
27. 27.Which of these numbers is divisible by 9? (a) 1323 (b) 1324 (c) 1321 (d) 1335
28. 28.Which of these numbers is divisible by 4? (a) 341 (b) 346 (c) 339 (d) 340
29. 29.Which of these numbers is divisible by 4? (a) 2000 (b) 1997 (c) 2005 (d) 2002
30. 30.What is the units digit of 8^20? (a) 2 (b) 8 (c) 6 (d) 4
31. 31.Which of these numbers is divisible by 9? (a) 2068 (b) 2047 (c) 2052 (d) 2054
32. 32.What is the units digit of 2^50? (a) 8 (b) 2 (c) 6 (d) 4
33. 33.Which of these numbers is divisible by 11? (a) 2575 (b) 2563 (c) 2565 (d) 2554
34. 34.Which of these numbers is divisible by 8? (a) 5918 (b) 5932 (c) 5924 (d) 5920
35. 35.What is the units digit of 3^20? (a) 3 (b) 9 (c) 1 (d) 7
36. 36.What is the units digit of 9^45? (a) 8 (b) 7 (c) 1 (d) 9
37. 37.What is the units digit of 4^56? (a) 7 (b) 5 (c) 4 (d) 6
38. 38.What is the units digit of 4^20? (a) 4 (b) 6 (c) 7 (d) 5
39. 39.What is the units digit of 4^24? (a) 7 (b) 5 (c) 4 (d) 6
40. 40.What is the smallest number you must add to 157 to make it divisible by 3? (a) 1 (b) 2

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