Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 594 · 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 123 to make it divisible by 11? (a) 8 (b) 10 (c) 9 (d) 2
2. 2.Which of these numbers is divisible by 6? (a) 4079 (b) 4087 (c) 4080 (d) 4085
3. 3.What is the smallest number you must add to 315 to make it divisible by 6? (a) 4 (b) 3 (c) 5 (d) 2
4. 4.What is the smallest number you must add to 109 to make it divisible by 8? (a) 2 (b) 4 (c) 5 (d) 3
5. 5.What is the smallest number you must add to 434 to make it divisible by 11? (a) 7 (b) 6 (c) 8 (d) 5
6. 6.Which of these numbers is divisible by 9? (a) 907 (b) 891 (c) 889 (d) 893
7. 7.What is the units digit of 3^41? (a) 7 (b) 9 (c) 3 (d) 1
8. 8.Which of these numbers is divisible by 3? (a) 163 (b) 161 (c) 159 (d) 157
9. 9.Which of these numbers is divisible by 11? (a) 9592 (b) 9583 (c) 9595 (d) 9610
10. 10.What is the remainder when 207 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
11. 11.Which of these numbers is divisible by 3? (a) 1376 (b) 1378 (c) 1381 (d) 1377
12. 12.Which of these numbers is divisible by 8? (a) 2461 (b) 2450 (c) 2471 (d) 2456
13. 13.Which of these numbers is divisible by 3? (a) 2307 (b) 2309 (c) 2311 (d) 2306
14. 14.What is the smallest number you must add to 373 to make it divisible by 7? (a) 6 (b) 2 (c) 4 (d) 5
15. 15.Which of these numbers is divisible by 8? (a) 5687 (b) 5689 (c) 5673 (d) 5680
16. 16.What is the smallest number you must add to 99 to make it divisible by 7? (a) 4 (b) 6 (c) 1 (d) 5
17. 17.What is the remainder when 156 is divided by 5? (a) 0 (b) 2 (c) 1 (d) 4
18. 18.What is the units digit of 2^55? (a) 2 (b) 8 (c) 6 (d) 4
19. 19.What is the smallest number you must add to 448 to make it divisible by 10? (a) 8 (b) 2 (c) 3 (d) 1
20. 20.What is the smallest number you must add to 281 to make it divisible by 12? (a) 7 (b) 5 (c) 8 (d) 6
21. 21.What is the remainder when 321 is divided by 8? (a) 0 (b) 7 (c) 2 (d) 1
22. 22.Which of these numbers is divisible by 4? (a) 1593 (b) 1590 (c) 1592 (d) 1599
23. 23.What is the units digit of 9^18? (a) 2 (b) 9 (c) 0 (d) 1
24. 24.Which of these numbers is divisible by 11? (a) 2244 (b) 2249 (c) 2264 (d) 2237
25. 25.Which of these numbers is divisible by 3? (a) 300 (b) 301 (c) 304 (d) 298
26. 26.Which of these numbers is divisible by 4? (a) 1775 (b) 1769 (c) 1779 (d) 1772
27. 27.What is the units digit of 8^12? (a) 2 (b) 6 (c) 8 (d) 4
28. 28.What is the remainder when 467 is divided by 7? (a) 4 (b) 5 (c) 6 (d) 2
29. 29.What is the remainder when 685 is divided by 10? (a) 5 (b) 7 (c) 4 (d) 6
30. 30.What is the remainder when 178 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
31. 31.Which of these numbers is divisible by 8? (a) 5436 (b) 5421 (c) 5424 (d) 5428
32. 32.What is the remainder when 145 is divided by 3? (a) 2 (b) 1 (c) 0
33. 33.Which of these numbers is divisible by 9? (a) 391 (b) 399 (c) 387 (d) 386
34. 34.What is the smallest number you must add to 577 to make it divisible by 11? (a) 6 (b) 8 (c) 5 (d) 7
35. 35.What is the remainder when 757 is divided by 10? (a) 8 (b) 7 (c) 6 (d) 3
36. 36.Which of these numbers is divisible by 11? (a) 1417 (b) 1437 (c) 1419 (d) 1429
37. 37.What is the units digit of 2^43? (a) 2 (b) 6 (c) 8 (d) 4
38. 38.What is the smallest number you must add to 56 to make it divisible by 3? (a) 2 (b) 1
39. 39.What is the remainder when 86 is divided by 7? (a) 2 (b) 1 (c) 3 (d) 5
40. 40.What is the units digit of 9^19? (a) 9 (b) 8 (c) 7 (d) 1

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