Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 988 · 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 58 to make it divisible by 3? (a) 1 (b) 2
2. 2.What is the smallest number you must add to 721 to make it divisible by 9? (a) 6 (b) 7 (c) 1 (d) 8
3. 3.What is the units digit of 4^56? (a) 5 (b) 7 (c) 6 (d) 4
4. 4.What is the remainder when 235 is divided by 3? (a) 0 (b) 2 (c) 1
5. 5.What is the units digit of 2^57? (a) 6 (b) 8 (c) 4 (d) 2
6. 6.What is the remainder when 602 is divided by 8? (a) 1 (b) 2 (c) 6 (d) 3
7. 7.What is the units digit of 8^23? (a) 2 (b) 8 (c) 6 (d) 4
8. 8.Which of these numbers is divisible by 9? (a) 360 (b) 353 (c) 364 (d) 372
9. 9.What is the smallest number you must add to 373 to make it divisible by 11? (a) 2 (b) 1 (c) 10 (d) 3
10. 10.What is the remainder when 238 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
11. 11.Which of these numbers is divisible by 4? (a) 3263 (b) 3254 (c) 3259 (d) 3256
12. 12.What is the remainder when 223 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
13. 13.What is the units digit of 8^43? (a) 8 (b) 6 (c) 2 (d) 4
14. 14.What is the remainder when 193 is divided by 3? (a) 1 (b) 2 (c) 0
15. 15.Which of these numbers is divisible by 6? (a) 4104 (b) 4113 (c) 4108 (d) 4099
16. 16.What is the units digit of 2^54? (a) 4 (b) 2 (c) 8 (d) 6
17. 17.What is the smallest number you must add to 262 to make it divisible by 10? (a) 2 (b) 9 (c) 7 (d) 8
18. 18.What is the smallest number you must add to 411 to make it divisible by 7? (a) 3 (b) 2 (c) 1 (d) 5
19. 19.What is the units digit of 2^41? (a) 2 (b) 8 (c) 6 (d) 4
20. 20.What is the units digit of 4^19? (a) 6 (b) 4 (c) 5 (d) 3
21. 21.Which of these numbers is divisible by 9? (a) 7833 (b) 7823 (c) 7830 (d) 7842
22. 22.Which of these numbers is divisible by 9? (a) 5661 (b) 5660 (c) 5678 (d) 5663
23. 23.Which of these numbers is divisible by 6? (a) 498 (b) 507 (c) 499 (d) 495
24. 24.What is the smallest number you must add to 442 to make it divisible by 8? (a) 5 (b) 7 (c) 2 (d) 6
25. 25.Which of these numbers is divisible by 9? (a) 2584 (b) 2582 (c) 2583 (d) 2599
26. 26.Which of these numbers is divisible by 3? (a) 2323 (b) 2319 (c) 2318 (d) 2321
27. 27.What is the remainder when 843 is divided by 11? (a) 7 (b) 4 (c) 6 (d) 8
28. 28.What is the smallest number you must add to 783 to make it divisible by 10? (a) 7 (b) 3 (c) 8 (d) 6
29. 29.What is the remainder when 116 is divided by 11? (a) 8 (b) 6 (c) 7 (d) 5
30. 30.What is the remainder when 291 is divided by 6? (a) 4 (b) 5 (c) 3 (d) 2
31. 31.What is the remainder when 37 is divided by 3? (a) 0 (b) 1 (c) 2
32. 32.Which of these numbers is divisible by 9? (a) 2003 (b) 2008 (c) 1998 (d) 1992
33. 33.What is the smallest number you must add to 277 to make it divisible by 8? (a) 2 (b) 3 (c) 5 (d) 4
34. 34.What is the units digit of 2^14? (a) 2 (b) 4 (c) 6 (d) 8
35. 35.What is the smallest number you must add to 323 to make it divisible by 9? (a) 2 (b) 3 (c) 1 (d) 8
36. 36.What is the remainder when 458 is divided by 12? (a) 10 (b) 1 (c) 3 (d) 2
37. 37.Which of these numbers is divisible by 4? (a) 1112 (b) 1113 (c) 1110 (d) 1118
38. 38.What is the smallest number you must add to 260 to make it divisible by 6? (a) 3 (b) 4 (c) 2 (d) 5
39. 39.What is the remainder when 316 is divided by 9? (a) 2 (b) 1 (c) 8 (d) 0
40. 40.What is the units digit of 2^38? (a) 6 (b) 8 (c) 4 (d) 2

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