Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 521 · 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 units digit of 4^33? (a) 3 (b) 6 (c) 5 (d) 4
2. 2.What is the smallest number you must add to 578 to make it divisible by 11? (a) 5 (b) 7 (c) 4 (d) 6
3. 3.What is the smallest number you must add to 354 to make it divisible by 8? (a) 6 (b) 5 (c) 7 (d) 2
4. 4.What is the units digit of 2^55? (a) 2 (b) 4 (c) 8 (d) 6
5. 5.Which of these numbers is divisible by 11? (a) 5168 (b) 5153 (c) 5172 (d) 5159
6. 6.What is the units digit of 2^41? (a) 4 (b) 2 (c) 6 (d) 8
7. 7.What is the remainder when 169 is divided by 3? (a) 0 (b) 1 (c) 2
8. 8.What is the units digit of 4^17? (a) 3 (b) 4 (c) 6 (d) 5
9. 9.What is the remainder when 260 is divided by 12? (a) 4 (b) 7 (c) 9 (d) 8
10. 10.Which of these numbers is divisible by 9? (a) 765 (b) 772 (c) 782 (d) 758
11. 11.Which of these numbers is divisible by 9? (a) 6266 (b) 6288 (c) 6274 (d) 6273
12. 12.What is the units digit of 2^56? (a) 2 (b) 4 (c) 8 (d) 6
13. 13.What is the units digit of 3^31? (a) 3 (b) 1 (c) 9 (d) 7
14. 14.What is the units digit of 4^31? (a) 6 (b) 5 (c) 3 (d) 4
15. 15.What is the units digit of 3^10? (a) 3 (b) 7 (c) 1 (d) 9
16. 16.What is the smallest number you must add to 447 to make it divisible by 10? (a) 3 (b) 7 (c) 4 (d) 2
17. 17.Which of these numbers is divisible by 4? (a) 1902 (b) 1906 (c) 1898 (d) 1900
18. 18.Which of these numbers is divisible by 3? (a) 1573 (b) 1572 (c) 1576 (d) 1571
19. 19.Which of these numbers is divisible by 6? (a) 4709 (b) 4715 (c) 4718 (d) 4710
20. 20.What is the units digit of 8^50? (a) 4 (b) 2 (c) 8 (d) 6
21. 21.What is the smallest number you must add to 64 to make it divisible by 6? (a) 2 (b) 1 (c) 3 (d) 4
22. 22.What is the units digit of 2^23? (a) 8 (b) 2 (c) 4 (d) 6
23. 23.Which of these numbers is divisible by 11? (a) 7617 (b) 7630 (c) 7609 (d) 7612
24. 24.What is the remainder when 190 is divided by 9? (a) 2 (b) 1 (c) 8 (d) 0
25. 25.What is the units digit of 7^13? (a) 7 (b) 1 (c) 9 (d) 3
26. 26.Which of these numbers is divisible by 8? (a) 3355 (b) 3375 (c) 3360 (d) 3366
27. 27.Which of these numbers is divisible by 3? (a) 428 (b) 429 (c) 431 (d) 434
28. 28.What is the remainder when 278 is divided by 12? (a) 3 (b) 10 (c) 1 (d) 2
29. 29.What is the remainder when 724 is divided by 9? (a) 5 (b) 6 (c) 3 (d) 4
30. 30.What is the units digit of 2^29? (a) 4 (b) 2 (c) 6 (d) 8
31. 31.What is the units digit of 8^21? (a) 4 (b) 6 (c) 8 (d) 2
32. 32.What is the smallest number you must add to 394 to make it divisible by 8? (a) 5 (b) 2 (c) 7 (d) 6
33. 33.What is the remainder when 109 is divided by 3? (a) 0 (b) 2 (c) 1
34. 34.What is the remainder when 668 is divided by 10? (a) 2 (b) 9 (c) 8 (d) 7
35. 35.What is the smallest number you must add to 253 to make it divisible by 6? (a) 3 (b) 1 (c) 5 (d) 4
36. 36.Which of these numbers is divisible by 6? (a) 1701 (b) 1707 (c) 1713 (d) 1704
37. 37.What is the remainder when 525 is divided by 8? (a) 6 (b) 4 (c) 3 (d) 5
38. 38.Which of these numbers is divisible by 3? (a) 1268 (b) 1270 (c) 1273 (d) 1269
39. 39.What is the remainder when 211 is divided by 6? (a) 0 (b) 2 (c) 1 (d) 5
40. 40.What is the remainder when 56 is divided by 3? (a) 1 (b) 0 (c) 2

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