Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 757 · 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) 1841 (b) 1839 (c) 1836 (d) 1833
2. 2.Which of these numbers is divisible by 9? (a) 2743 (b) 2736 (c) 2749 (d) 2729
3. 3.What is the smallest number you must add to 688 to make it divisible by 10? (a) 1 (b) 8 (c) 3 (d) 2
4. 4.What is the smallest number you must add to 341 to make it divisible by 7? (a) 5 (b) 2 (c) 3 (d) 1
5. 5.Which of these numbers is divisible by 6? (a) 1435 (b) 1444 (c) 1434 (d) 1429
6. 6.Which of these numbers is divisible by 11? (a) 2088 (b) 2079 (c) 2076 (d) 2096
7. 7.What is the smallest number you must add to 147 to make it divisible by 9? (a) 3 (b) 5 (c) 7 (d) 6
8. 8.What is the remainder when 746 is divided by 11? (a) 10 (b) 2 (c) 8 (d) 9
9. 9.What is the units digit of 2^60? (a) 8 (b) 2 (c) 4 (d) 6
10. 10.What is the remainder when 359 is divided by 8? (a) 6 (b) 1 (c) 5 (d) 7
11. 11.What is the units digit of 7^15? (a) 3 (b) 1 (c) 9 (d) 7
12. 12.What is the units digit of 7^44? (a) 7 (b) 9 (c) 3 (d) 1
13. 13.What is the remainder when 156 is divided by 11? (a) 1 (b) 2 (c) 3 (d) 9
14. 14.What is the units digit of 2^30? (a) 2 (b) 8 (c) 6 (d) 4
15. 15.What is the units digit of 8^59? (a) 2 (b) 8 (c) 6 (d) 4
16. 16.Which of these numbers is divisible by 8? (a) 5010 (b) 5021 (c) 5016 (d) 5030
17. 17.What is the smallest number you must add to 509 to make it divisible by 8? (a) 3 (b) 2 (c) 4 (d) 5
18. 18.What is the smallest number you must add to 579 to make it divisible by 9? (a) 3 (b) 6 (c) 5 (d) 7
19. 19.What is the remainder when 431 is divided by 8? (a) 1 (b) 6 (c) 5 (d) 7
20. 20.What is the units digit of 7^21? (a) 3 (b) 7 (c) 1 (d) 9
21. 21.What is the remainder when 316 is divided by 6? (a) 5 (b) 3 (c) 2 (d) 4
22. 22.Which of these numbers is divisible by 6? (a) 2548 (b) 2555 (c) 2544 (d) 2540
23. 23.What is the units digit of 3^22? (a) 1 (b) 9 (c) 7 (d) 3
24. 24.What is the remainder when 116 is divided by 11? (a) 6 (b) 8 (c) 5 (d) 7
25. 25.What is the smallest number you must add to 53 to make it divisible by 3? (a) 1 (b) 2
26. 26.What is the remainder when 628 is divided by 8? (a) 3 (b) 6 (c) 4 (d) 5
27. 27.Which of these numbers is divisible by 6? (a) 995 (b) 996 (c) 1000 (d) 1007
28. 28.Which of these numbers is divisible by 11? (a) 1324 (b) 1319 (c) 1320 (d) 1337
29. 29.What is the remainder when 158 is divided by 3? (a) 1 (b) 0 (c) 2
30. 30.What is the smallest number you must add to 67 to make it divisible by 6? (a) 1 (b) 5 (c) 4 (d) 3
31. 31.Which of these numbers is divisible by 6? (a) 3055 (b) 3051 (c) 3048 (d) 3043
32. 32.What is the smallest number you must add to 149 to make it divisible by 6? (a) 1 (b) 2 (c) 3 (d) 5
33. 33.Which of these numbers is divisible by 9? (a) 1555 (b) 1565 (c) 1568 (d) 1557
34. 34.Which of these numbers is divisible by 9? (a) 6285 (b) 6291 (c) 6307 (d) 6296
35. 35.Which of these numbers is divisible by 4? (a) 235 (b) 237 (c) 230 (d) 232
36. 36.Which of these numbers is divisible by 9? (a) 943 (b) 946 (c) 930 (d) 936
37. 37.What is the remainder when 116 is divided by 7? (a) 5 (b) 3 (c) 6 (d) 4
38. 38.What is the remainder when 371 is divided by 5? (a) 4 (b) 1 (c) 0 (d) 2
39. 39.What is the units digit of 9^18? (a) 2 (b) 1 (c) 0 (d) 9
40. 40.Which of these numbers is divisible by 11? (a) 8822 (b) 8820 (c) 8826 (d) 8841

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