Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 80 · 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) 2217 (b) 2215 (c) 2221 (d) 2216
2. 2.What is the smallest number you must add to 215 to make it divisible by 9? (a) 8 (b) 3 (c) 2 (d) 1
3. 3.What is the smallest number you must add to 770 to make it divisible by 12? (a) 9 (b) 11 (c) 10 (d) 2
4. 4.Which of these numbers is divisible by 11? (a) 1745 (b) 1720 (c) 1727 (d) 1734
5. 5.What is the smallest number you must add to 314 to make it divisible by 12? (a) 9 (b) 2 (c) 11 (d) 10
6. 6.What is the remainder when 291 is divided by 12? (a) 9 (b) 2 (c) 3 (d) 4
7. 7.What is the units digit of 2^49? (a) 2 (b) 8 (c) 6 (d) 4
8. 8.What is the remainder when 401 is divided by 12? (a) 6 (b) 7 (c) 4 (d) 5
9. 9.Which of these numbers is divisible by 4? (a) 1748 (b) 1751 (c) 1747 (d) 1754
10. 10.What is the remainder when 67 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
11. 11.What is the remainder when 351 is divided by 11? (a) 9 (b) 10 (c) 1 (d) 8
12. 12.What is the remainder when 428 is divided by 11? (a) 9 (b) 1 (c) 10 (d) 8
13. 13.What is the units digit of 3^28? (a) 7 (b) 9 (c) 1 (d) 3
14. 14.What is the remainder when 137 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
15. 15.What is the units digit of 8^43? (a) 4 (b) 6 (c) 2 (d) 8
16. 16.What is the units digit of 9^49? (a) 7 (b) 8 (c) 9 (d) 1
17. 17.What is the remainder when 796 is divided by 12? (a) 5 (b) 8 (c) 4 (d) 3
18. 18.What is the units digit of 7^38? (a) 7 (b) 3 (c) 1 (d) 9
19. 19.What is the units digit of 4^17? (a) 3 (b) 5 (c) 6 (d) 4
20. 20.What is the units digit of 7^50? (a) 9 (b) 3 (c) 1 (d) 7
21. 21.Which of these numbers is divisible by 3? (a) 377 (b) 370 (c) 374 (d) 372
22. 22.What is the remainder when 163 is divided by 10? (a) 7 (b) 4 (c) 3 (d) 2
23. 23.Which of these numbers is divisible by 3? (a) 1808 (b) 1811 (c) 1806 (d) 1804
24. 24.What is the smallest number you must add to 73 to make it divisible by 7? (a) 4 (b) 6 (c) 5 (d) 3
25. 25.What is the units digit of 7^44? (a) 3 (b) 9 (c) 7 (d) 1
26. 26.What is the units digit of 9^33? (a) 8 (b) 1 (c) 7 (d) 9
27. 27.What is the units digit of 7^12? (a) 3 (b) 1 (c) 9 (d) 7
28. 28.What is the smallest number you must add to 111 to make it divisible by 4? (a) 1 (b) 2 (c) 3
29. 29.Which of these numbers is divisible by 8? (a) 1884 (b) 1870 (c) 1872 (d) 1877
30. 30.What is the remainder when 715 is divided by 10? (a) 4 (b) 6 (c) 5 (d) 7
31. 31.Which of these numbers is divisible by 3? (a) 2067 (b) 2068 (c) 2065 (d) 2072
32. 32.What is the remainder when 744 is divided by 11? (a) 6 (b) 7 (c) 8 (d) 4
33. 33.Which of these numbers is divisible by 4? (a) 3055 (b) 3052 (c) 3057 (d) 3051
34. 34.Which of these numbers is divisible by 4? (a) 877 (b) 876 (c) 881 (d) 875
35. 35.What is the smallest number you must add to 194 to make it divisible by 3? (a) 1 (b) 2
36. 36.What is the smallest number you must add to 884 to make it divisible by 11? (a) 4 (b) 7 (c) 6 (d) 8
37. 37.What is the smallest number you must add to 68 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
38. 38.What is the remainder when 235 is divided by 3? (a) 2 (b) 0 (c) 1
39. 39.What is the units digit of 4^14? (a) 5 (b) 6 (c) 7 (d) 4
40. 40.Which of these numbers is divisible by 4? (a) 2994 (b) 2992 (c) 2998 (d) 2991

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