Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 773 · 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 remainder when 126 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 0
2. 2.What is the units digit of 2^22? (a) 2 (b) 6 (c) 8 (d) 4
3. 3.What is the units digit of 9^32? (a) 0 (b) 2 (c) 9 (d) 1
4. 4.Which of these numbers is divisible by 6? (a) 514 (b) 508 (c) 503 (d) 504
5. 5.Which of these numbers is divisible by 6? (a) 4417 (b) 4406 (c) 4415 (d) 4410
6. 6.What is the remainder when 318 is divided by 7? (a) 4 (b) 3 (c) 5 (d) 2
7. 7.What is the remainder when 129 is divided by 6? (a) 3 (b) 5 (c) 4 (d) 2
8. 8.What is the units digit of 4^16? (a) 6 (b) 5 (c) 4 (d) 7
9. 9.Which of these numbers is divisible by 4? (a) 1471 (b) 1473 (c) 1472 (d) 1479
10. 10.Which of these numbers is divisible by 3? (a) 1651 (b) 1645 (c) 1647 (d) 1649
11. 11.Which of these numbers is divisible by 9? (a) 6104 (b) 6123 (c) 6119 (d) 6111
12. 12.What is the units digit of 9^26? (a) 2 (b) 0 (c) 9 (d) 1
13. 13.Which of these numbers is divisible by 11? (a) 4047 (b) 4051 (c) 4064 (d) 4048
14. 14.Which of these numbers is divisible by 3? (a) 2444 (b) 2439 (c) 2438 (d) 2440
15. 15.What is the remainder when 299 is divided by 6? (a) 1 (b) 5 (c) 4 (d) 3
16. 16.Which of these numbers is divisible by 8? (a) 663 (b) 647 (c) 648 (d) 654
17. 17.Which of these numbers is divisible by 11? (a) 6412 (b) 6416 (c) 6413 (d) 6427
18. 18.What is the smallest number you must add to 51 to make it divisible by 4? (a) 1 (b) 3 (c) 2
19. 19.What is the remainder when 162 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
20. 20.Which of these numbers is divisible by 6? (a) 4857 (b) 4864 (c) 4854 (d) 4850
21. 21.What is the units digit of 7^44? (a) 7 (b) 1 (c) 3 (d) 9
22. 22.What is the units digit of 9^24? (a) 9 (b) 1 (c) 2 (d) 0
23. 23.Which of these numbers is divisible by 3? (a) 1088 (b) 1091 (c) 1085 (d) 1086
24. 24.Which of these numbers is divisible by 3? (a) 1456 (b) 1462 (c) 1458 (d) 1460
25. 25.Which of these numbers is divisible by 6? (a) 749 (b) 734 (c) 738 (d) 741
26. 26.Which of these numbers is divisible by 4? (a) 180 (b) 181 (c) 177 (d) 186
27. 27.Which of these numbers is divisible by 3? (a) 831 (b) 829 (c) 832 (d) 835
28. 28.What is the remainder when 450 is divided by 8? (a) 6 (b) 1 (c) 3 (d) 2
29. 29.What is the remainder when 104 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
30. 30.What is the smallest number you must add to 191 to make it divisible by 8? (a) 2 (b) 1 (c) 3 (d) 7
31. 31.What is the remainder when 318 is divided by 9? (a) 2 (b) 3 (c) 6 (d) 4
32. 32.What is the units digit of 3^19? (a) 3 (b) 7 (c) 1 (d) 9
33. 33.Which of these numbers is divisible by 8? (a) 6516 (b) 6504 (c) 6503 (d) 6511
34. 34.What is the smallest number you must add to 241 to make it divisible by 3? (a) 2 (b) 1
35. 35.Which of these numbers is divisible by 9? (a) 2362 (b) 2358 (c) 2353 (d) 2375
36. 36.What is the units digit of 2^39? (a) 2 (b) 6 (c) 4 (d) 8
37. 37.Which of these numbers is divisible by 4? (a) 1649 (b) 1644 (c) 1642 (d) 1647
38. 38.Which of these numbers is divisible by 3? (a) 224 (b) 226 (c) 220 (d) 222
39. 39.Which of these numbers is divisible by 8? (a) 1111 (b) 1099 (c) 1096 (d) 1090
40. 40.What is the smallest number you must add to 464 to make it divisible by 7? (a) 5 (b) 2 (c) 4 (d) 6

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