Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 120 · 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 65 to make it divisible by 6? (a) 2 (b) 5 (c) 1 (d) 3
2. 2.What is the smallest number you must add to 79 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 1
3. 3.What is the remainder when 203 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
4. 4.Which of these numbers is divisible by 9? (a) 2629 (b) 2644 (c) 2637 (d) 2651
5. 5.What is the units digit of 2^30? (a) 6 (b) 2 (c) 4 (d) 8
6. 6.What is the smallest number you must add to 321 to make it divisible by 4? (a) 3 (b) 1 (c) 2
7. 7.Which of these numbers is divisible by 4? (a) 1483 (b) 1474 (c) 1476 (d) 1478
8. 8.Which of these numbers is divisible by 4? (a) 2159 (b) 2162 (c) 2160 (d) 2167
9. 9.What is the remainder when 158 is divided by 6? (a) 2 (b) 1 (c) 4 (d) 3
10. 10.What is the smallest number you must add to 381 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
11. 11.What is the remainder when 756 is divided by 11? (a) 7 (b) 9 (c) 8 (d) 3
12. 12.What is the units digit of 9^13? (a) 9 (b) 1 (c) 7 (d) 8
13. 13.What is the units digit of 9^33? (a) 7 (b) 8 (c) 9 (d) 1
14. 14.Which of these numbers is divisible by 9? (a) 7429 (b) 7445 (c) 7434 (d) 7436
15. 15.What is the remainder when 150 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
16. 16.What is the units digit of 2^46? (a) 6 (b) 2 (c) 8 (d) 4
17. 17.Which of these numbers is divisible by 3? (a) 701 (b) 697 (c) 696 (d) 695
18. 18.What is the units digit of 4^30? (a) 6 (b) 4 (c) 5 (d) 7
19. 19.What is the units digit of 4^41? (a) 6 (b) 5 (c) 3 (d) 4
20. 20.Which of these numbers is divisible by 8? (a) 874 (b) 871 (c) 872 (d) 881
21. 21.Which of these numbers is divisible by 9? (a) 4344 (b) 4335 (c) 4329 (d) 4323
22. 22.What is the remainder when 219 is divided by 7? (a) 1 (b) 5 (c) 3 (d) 2
23. 23.Which of these numbers is divisible by 9? (a) 6326 (b) 6332 (c) 6327 (d) 6342
24. 24.What is the remainder when 446 is divided by 7? (a) 4 (b) 2 (c) 6 (d) 5
25. 25.What is the smallest number you must add to 329 to make it divisible by 8? (a) 6 (b) 1 (c) 5 (d) 7
26. 26.What is the units digit of 2^44? (a) 6 (b) 8 (c) 2 (d) 4
27. 27.What is the remainder when 638 is divided by 8? (a) 6 (b) 5 (c) 7 (d) 2
28. 28.What is the remainder when 594 is divided by 8? (a) 3 (b) 1 (c) 2 (d) 6
29. 29.What is the units digit of 2^23? (a) 2 (b) 4 (c) 6 (d) 8
30. 30.What is the smallest number you must add to 570 to make it divisible by 9? (a) 5 (b) 7 (c) 6 (d) 3
31. 31.Which of these numbers is divisible by 6? (a) 3249 (b) 3237 (c) 3240 (d) 3245
32. 32.What is the smallest number you must add to 603 to make it divisible by 8? (a) 3 (b) 4 (c) 6 (d) 5
33. 33.What is the smallest number you must add to 214 to make it divisible by 3? (a) 2 (b) 1
34. 34.Which of these numbers is divisible by 4? (a) 1599 (b) 1596 (c) 1595 (d) 1603
35. 35.What is the remainder when 315 is divided by 11? (a) 8 (b) 4 (c) 6 (d) 7
36. 36.What is the units digit of 3^60? (a) 7 (b) 3 (c) 1 (d) 9
37. 37.What is the remainder when 53 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
38. 38.Which of these numbers is divisible by 4? (a) 1096 (b) 1103 (c) 1099 (d) 1094
39. 39.Which of these numbers is divisible by 3? (a) 1201 (b) 1204 (c) 1207 (d) 1203
40. 40.Which of these numbers is divisible by 9? (a) 7502 (b) 7493 (c) 7497 (d) 7507

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