Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 221 · 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) 5720 (b) 5732 (c) 5713 (d) 5725
2. 2.What is the units digit of 9^29? (a) 1 (b) 7 (c) 9 (d) 8
3. 3.Which of these numbers is divisible by 6? (a) 5194 (b) 5183 (c) 5189 (d) 5184
4. 4.What is the smallest number you must add to 138 to make it divisible by 10? (a) 1 (b) 2 (c) 3 (d) 8
5. 5.Which of these numbers is divisible by 6? (a) 430 (b) 420 (c) 416 (d) 424
6. 6.Which of these numbers is divisible by 3? (a) 2387 (b) 2392 (c) 2390 (d) 2388
7. 7.What is the remainder when 250 is divided by 7? (a) 5 (b) 2 (c) 4 (d) 6
8. 8.What is the remainder when 383 is divided by 6? (a) 3 (b) 5 (c) 1 (d) 4
9. 9.What is the units digit of 2^59? (a) 2 (b) 6 (c) 8 (d) 4
10. 10.What is the units digit of 2^45? (a) 4 (b) 6 (c) 2 (d) 8
11. 11.What is the units digit of 3^42? (a) 7 (b) 1 (c) 9 (d) 3
12. 12.Which of these numbers is divisible by 3? (a) 2498 (b) 2499 (c) 2503 (d) 2500
13. 13.What is the units digit of 3^55? (a) 7 (b) 9 (c) 3 (d) 1
14. 14.What is the units digit of 3^21? (a) 9 (b) 1 (c) 7 (d) 3
15. 15.What is the units digit of 9^19? (a) 9 (b) 8 (c) 1 (d) 7
16. 16.What is the units digit of 2^29? (a) 8 (b) 2 (c) 4 (d) 6
17. 17.Which of these numbers is divisible by 8? (a) 3376 (b) 3381 (c) 3387 (d) 3373
18. 18.What is the units digit of 4^59? (a) 5 (b) 4 (c) 6 (d) 3
19. 19.What is the remainder when 266 is divided by 5? (a) 4 (b) 0 (c) 1 (d) 2
20. 20.What is the smallest number you must add to 355 to make it divisible by 12? (a) 6 (b) 7 (c) 4 (d) 5
21. 21.What is the remainder when 173 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
22. 22.What is the remainder when 621 is divided by 8? (a) 3 (b) 5 (c) 6 (d) 4
23. 23.What is the smallest number you must add to 697 to make it divisible by 11? (a) 7 (b) 8 (c) 4 (d) 6
24. 24.What is the remainder when 320 is divided by 6? (a) 4 (b) 3 (c) 2 (d) 1
25. 25.Which of these numbers is divisible by 6? (a) 2028 (b) 2035 (c) 2027 (d) 2030
26. 26.What is the remainder when 508 is divided by 8? (a) 5 (b) 6 (c) 3 (d) 4
27. 27.What is the smallest number you must add to 861 to make it divisible by 11? (a) 3 (b) 7 (c) 8 (d) 9
28. 28.What is the smallest number you must add to 409 to make it divisible by 8? (a) 6 (b) 7 (c) 5 (d) 1
29. 29.Which of these numbers is divisible by 3? (a) 2676 (b) 2674 (c) 2677 (d) 2681
30. 30.What is the smallest number you must add to 123 to make it divisible by 12? (a) 9 (b) 8 (c) 3 (d) 10
31. 31.What is the remainder when 69 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
32. 32.Which of these numbers is divisible by 3? (a) 387 (b) 389 (c) 392 (d) 385
33. 33.What is the smallest number you must add to 444 to make it divisible by 9? (a) 7 (b) 5 (c) 3 (d) 6
34. 34.Which of these numbers is divisible by 6? (a) 4621 (b) 4615 (c) 4620 (d) 4631
35. 35.What is the units digit of 9^31? (a) 7 (b) 9 (c) 1 (d) 8
36. 36.Which of these numbers is divisible by 11? (a) 5293 (b) 5305 (c) 5291 (d) 5287
37. 37.Which of these numbers is divisible by 4? (a) 925 (b) 929 (c) 924 (d) 923
38. 38.What is the remainder when 448 is divided by 10? (a) 2 (b) 8 (c) 9 (d) 7
39. 39.What is the smallest number you must add to 258 to make it divisible by 4? (a) 2 (b) 3 (c) 1
40. 40.What is the smallest number you must add to 364 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3

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