Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 817 · 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 9^16? (a) 9 (b) 0 (c) 1 (d) 2
2. 2.Which of these numbers is divisible by 4? (a) 2839 (b) 2843 (c) 2845 (d) 2840
3. 3.Which of these numbers is divisible by 3? (a) 1021 (b) 1025 (c) 1023 (d) 1027
4. 4.What is the remainder when 179 is divided by 9? (a) 6 (b) 1 (c) 7 (d) 8
5. 5.What is the smallest number you must add to 146 to make it divisible by 3? (a) 2 (b) 1
6. 6.What is the smallest number you must add to 598 to make it divisible by 8? (a) 2 (b) 1 (c) 3 (d) 6
7. 7.What is the smallest number you must add to 128 to make it divisible by 3? (a) 1 (b) 2
8. 8.What is the units digit of 3^15? (a) 9 (b) 7 (c) 1 (d) 3
9. 9.Which of these numbers is divisible by 11? (a) 1158 (b) 1149 (c) 1167 (d) 1155
10. 10.What is the smallest number you must add to 732 to make it divisible by 10? (a) 2 (b) 7 (c) 8 (d) 9
11. 11.What is the remainder when 261 is divided by 7? (a) 3 (b) 1 (c) 2 (d) 5
12. 12.Which of these numbers is divisible by 6? (a) 306 (b) 301 (c) 309 (d) 314
13. 13.Which of these numbers is divisible by 9? (a) 7235 (b) 7212 (c) 7218 (d) 7221
14. 14.Which of these numbers is divisible by 4? (a) 401 (b) 398 (c) 396 (d) 395
15. 15.Which of these numbers is divisible by 4? (a) 716 (b) 723 (c) 719 (d) 713
16. 16.Which of these numbers is divisible by 3? (a) 2000 (b) 1995 (c) 1994 (d) 1996
17. 17.What is the remainder when 707 is divided by 11? (a) 8 (b) 4 (c) 3 (d) 2
18. 18.What is the units digit of 8^23? (a) 8 (b) 4 (c) 2 (d) 6
19. 19.What is the remainder when 289 is divided by 9? (a) 0 (b) 2 (c) 1 (d) 8
20. 20.Which of these numbers is divisible by 4? (a) 1271 (b) 1274 (c) 1278 (d) 1272
21. 21.What is the smallest number you must add to 202 to make it divisible by 11? (a) 6 (b) 4 (c) 8 (d) 7
22. 22.Which of these numbers is divisible by 11? (a) 5600 (b) 5611 (c) 5599 (d) 5593
23. 23.Which of these numbers is divisible by 8? (a) 1406 (b) 1414 (c) 1397 (d) 1400
24. 24.What is the units digit of 2^44? (a) 2 (b) 8 (c) 6 (d) 4
25. 25.What is the smallest number you must add to 103 to make it divisible by 3? (a) 1 (b) 2
26. 26.What is the smallest number you must add to 327 to make it divisible by 10? (a) 3 (b) 2 (c) 7 (d) 4
27. 27.What is the remainder when 94 is divided by 9? (a) 3 (b) 5 (c) 6 (d) 4
28. 28.What is the remainder when 221 is divided by 3? (a) 1 (b) 2 (c) 0
29. 29.Which of these numbers is divisible by 11? (a) 583 (b) 588 (c) 595 (d) 578
30. 30.Which of these numbers is divisible by 9? (a) 4326 (b) 4314 (c) 4311 (d) 4306
31. 31.What is the remainder when 233 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
32. 32.What is the remainder when 322 is divided by 9? (a) 7 (b) 2 (c) 6 (d) 8
33. 33.What is the remainder when 254 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
34. 34.Which of these numbers is divisible by 4? (a) 2177 (b) 2173 (c) 2172 (d) 2171
35. 35.What is the units digit of 9^57? (a) 8 (b) 7 (c) 1 (d) 9
36. 36.Which of these numbers is divisible by 4? (a) 262 (b) 256 (c) 255 (d) 258
37. 37.What is the units digit of 2^33? (a) 4 (b) 2 (c) 8 (d) 6
38. 38.What is the remainder when 568 is divided by 10? (a) 2 (b) 8 (c) 7 (d) 9
39. 39.What is the units digit of 8^47? (a) 8 (b) 2 (c) 6 (d) 4
40. 40.What is the units digit of 7^39? (a) 3 (b) 7 (c) 9 (d) 1

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