Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 629 · 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 98 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
2. 2.Which of these numbers is divisible by 8? (a) 3756 (b) 3744 (c) 3751 (d) 3743
3. 3.What is the remainder when 93 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
4. 4.What is the units digit of 7^55? (a) 1 (b) 3 (c) 9 (d) 7
5. 5.What is the smallest number you must add to 558 to make it divisible by 8? (a) 2 (b) 1 (c) 3 (d) 6
6. 6.What is the smallest number you must add to 100 to make it divisible by 3? (a) 1 (b) 2
7. 7.Which of these numbers is divisible by 9? (a) 391 (b) 380 (c) 378 (d) 372
8. 8.Which of these numbers is divisible by 8? (a) 6669 (b) 6672 (c) 6686 (d) 6675
9. 9.What is the smallest number you must add to 575 to make it divisible by 11? (a) 9 (b) 3 (c) 8 (d) 7
10. 10.What is the smallest number you must add to 286 to make it divisible by 4? (a) 1 (b) 3 (c) 2
11. 11.What is the smallest number you must add to 64 to make it divisible by 3? (a) 2 (b) 1
12. 12.Which of these numbers is divisible by 3? (a) 1331 (b) 1326 (c) 1327 (d) 1325
13. 13.What is the remainder when 874 is divided by 11? (a) 6 (b) 7 (c) 4 (d) 5
14. 14.What is the units digit of 2^36? (a) 8 (b) 6 (c) 4 (d) 2
15. 15.What is the smallest number you must add to 235 to make it divisible by 4? (a) 3 (b) 2 (c) 1
16. 16.What is the units digit of 8^13? (a) 6 (b) 4 (c) 2 (d) 8
17. 17.Which of these numbers is divisible by 9? (a) 1989 (b) 2005 (c) 1988 (d) 1992
18. 18.What is the remainder when 227 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
19. 19.What is the smallest number you must add to 122 to make it divisible by 7? (a) 6 (b) 4 (c) 3 (d) 5
20. 20.Which of these numbers is divisible by 11? (a) 2145 (b) 2160 (c) 2146 (d) 2144
21. 21.What is the units digit of 2^28? (a) 6 (b) 4 (c) 2 (d) 8
22. 22.What is the units digit of 2^49? (a) 8 (b) 2 (c) 6 (d) 4
23. 23.What is the remainder when 729 is divided by 10? (a) 8 (b) 1 (c) 9 (d) 7
24. 24.What is the smallest number you must add to 559 to make it divisible by 12? (a) 6 (b) 5 (c) 7 (d) 4
25. 25.Which of these numbers is divisible by 4? (a) 2977 (b) 2980 (c) 2987 (d) 2983
26. 26.What is the remainder when 197 is divided by 3? (a) 2 (b) 0 (c) 1
27. 27.Which of these numbers is divisible by 6? (a) 1406 (b) 1418 (c) 1413 (d) 1410
28. 28.What is the units digit of 8^23? (a) 8 (b) 6 (c) 2 (d) 4
29. 29.Which of these numbers is divisible by 11? (a) 9053 (b) 9049 (c) 9071 (d) 9054
30. 30.What is the remainder when 560 is divided by 9? (a) 7 (b) 2 (c) 1 (d) 3
31. 31.Which of these numbers is divisible by 3? (a) 279 (b) 284 (c) 281 (d) 277
32. 32.Which of these numbers is divisible by 8? (a) 7132 (b) 7128 (c) 7121 (d) 7138
33. 33.Which of these numbers is divisible by 3? (a) 1442 (b) 1444 (c) 1447 (d) 1443
34. 34.What is the units digit of 2^17? (a) 4 (b) 8 (c) 6 (d) 2
35. 35.What is the units digit of 2^35? (a) 4 (b) 2 (c) 8 (d) 6
36. 36.What is the smallest number you must add to 157 to make it divisible by 9? (a) 4 (b) 5 (c) 7 (d) 6
37. 37.What is the smallest number you must add to 493 to make it divisible by 11? (a) 9 (b) 1 (c) 2 (d) 3
38. 38.What is the units digit of 2^10? (a) 4 (b) 8 (c) 2 (d) 6
39. 39.What is the smallest number you must add to 651 to make it divisible by 11? (a) 2 (b) 9 (c) 10 (d) 8
40. 40.What is the units digit of 3^15? (a) 1 (b) 7 (c) 3 (d) 9

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