Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 269 · 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 4? (a) 1737 (b) 1747 (c) 1741 (d) 1740
2. 2.What is the smallest number you must add to 98 to make it divisible by 3? (a) 2 (b) 1
3. 3.Which of these numbers is divisible by 3? (a) 1640 (b) 1633 (c) 1635 (d) 1637
4. 4.What is the remainder when 361 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 0
5. 5.What is the units digit of 9^37? (a) 7 (b) 9 (c) 1 (d) 8
6. 6.What is the remainder when 486 is divided by 7? (a) 3 (b) 5 (c) 4 (d) 2
7. 7.What is the units digit of 9^56? (a) 1 (b) 2 (c) 9 (d) 0
8. 8.Which of these numbers is divisible by 3? (a) 1867 (b) 1865 (c) 1861 (d) 1863
9. 9.Which of these numbers is divisible by 9? (a) 7544 (b) 7527 (c) 7533 (d) 7536
10. 10.What is the remainder when 253 is divided by 9? (a) 2 (b) 8 (c) 0 (d) 1
11. 11.What is the smallest number you must add to 158 to make it divisible by 7? (a) 5 (b) 4 (c) 2 (d) 3
12. 12.What is the remainder when 158 is divided by 9? (a) 6 (b) 7 (c) 4 (d) 5
13. 13.What is the remainder when 333 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
14. 14.Which of these numbers is divisible by 11? (a) 832 (b) 838 (c) 836 (d) 848
15. 15.What is the remainder when 493 is divided by 12? (a) 11 (b) 1 (c) 0 (d) 2
16. 16.What is the units digit of 3^50? (a) 1 (b) 3 (c) 9 (d) 7
17. 17.What is the remainder when 248 is divided by 10? (a) 8 (b) 2 (c) 9 (d) 7
18. 18.Which of these numbers is divisible by 9? (a) 4445 (b) 4428 (c) 4434 (d) 4421
19. 19.What is the smallest number you must add to 103 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
20. 20.Which of these numbers is divisible by 4? (a) 2174 (b) 2182 (c) 2176 (d) 2179
21. 21.What is the units digit of 4^23? (a) 3 (b) 6 (c) 4 (d) 5
22. 22.What is the units digit of 7^21? (a) 3 (b) 7 (c) 9 (d) 1
23. 23.Which of these numbers is divisible by 4? (a) 3312 (b) 3318 (c) 3313 (d) 3311
24. 24.What is the units digit of 2^20? (a) 6 (b) 4 (c) 8 (d) 2
25. 25.Which of these numbers is divisible by 3? (a) 527 (b) 523 (c) 525 (d) 529
26. 26.What is the units digit of 2^14? (a) 6 (b) 4 (c) 2 (d) 8
27. 27.Which of these numbers is divisible by 6? (a) 1094 (b) 1084 (c) 1089 (d) 1086
28. 28.Which of these numbers is divisible by 8? (a) 4381 (b) 4393 (c) 4384 (d) 4387
29. 29.What is the remainder when 345 is divided by 7? (a) 5 (b) 2 (c) 1 (d) 3
30. 30.What is the remainder when 203 is divided by 9? (a) 4 (b) 5 (c) 7 (d) 6
31. 31.What is the smallest number you must add to 309 to make it divisible by 4? (a) 2 (b) 3 (c) 1
32. 32.What is the smallest number you must add to 710 to make it divisible by 12? (a) 9 (b) 11 (c) 2 (d) 10
33. 33.What is the units digit of 2^19? (a) 8 (b) 6 (c) 2 (d) 4
34. 34.Which of these numbers is divisible by 8? (a) 5997 (b) 5988 (c) 5977 (d) 5984
35. 35.What is the smallest number you must add to 763 to make it divisible by 11? (a) 4 (b) 6 (c) 8 (d) 7
36. 36.What is the units digit of 8^23? (a) 8 (b) 4 (c) 6 (d) 2
37. 37.What is the smallest number you must add to 93 to make it divisible by 6? (a) 2 (b) 4 (c) 5 (d) 3
38. 38.Which of these numbers is divisible by 4? (a) 3190 (b) 3186 (c) 3184 (d) 3183
39. 39.What is the units digit of 9^39? (a) 9 (b) 7 (c) 1 (d) 8
40. 40.What is the smallest number you must add to 98 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4

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