Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 373 · 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 4^35? (a) 6 (b) 5 (c) 4 (d) 3
2. 2.What is the smallest number you must add to 277 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
3. 3.What is the remainder when 113 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
4. 4.What is the smallest number you must add to 224 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
5. 5.What is the units digit of 2^36? (a) 4 (b) 6 (c) 8 (d) 2
6. 6.What is the remainder when 367 is divided by 10? (a) 8 (b) 7 (c) 3 (d) 6
7. 7.What is the remainder when 543 is divided by 8? (a) 5 (b) 7 (c) 6 (d) 1
8. 8.What is the smallest number you must add to 86 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
9. 9.What is the smallest number you must add to 366 to make it divisible by 9? (a) 4 (b) 6 (c) 3 (d) 2
10. 10.What is the remainder when 210 is divided by 12? (a) 8 (b) 5 (c) 6 (d) 7
11. 11.What is the units digit of 8^17? (a) 4 (b) 2 (c) 6 (d) 8
12. 12.What is the units digit of 8^11? (a) 4 (b) 2 (c) 6 (d) 8
13. 13.Which of these numbers is divisible by 6? (a) 1022 (b) 1028 (c) 1034 (d) 1026
14. 14.What is the remainder when 537 is divided by 12? (a) 9 (b) 10 (c) 3 (d) 8
15. 15.Which of these numbers is divisible by 3? (a) 2559 (b) 2560 (c) 2564 (d) 2557
16. 16.What is the smallest number you must add to 275 to make it divisible by 8? (a) 3 (b) 4 (c) 6 (d) 5
17. 17.What is the remainder when 487 is divided by 8? (a) 6 (b) 7 (c) 5 (d) 1
18. 18.What is the smallest number you must add to 143 to make it divisible by 3? (a) 2 (b) 1
19. 19.Which of these numbers is divisible by 9? (a) 1362 (b) 1347 (c) 1351 (d) 1350
20. 20.Which of these numbers is divisible by 9? (a) 2785 (b) 2781 (c) 2791 (d) 2773
21. 21.What is the remainder when 491 is divided by 9? (a) 6 (b) 4 (c) 7 (d) 5
22. 22.Which of these numbers is divisible by 3? (a) 1730 (b) 1723 (c) 1727 (d) 1725
23. 23.Which of these numbers is divisible by 9? (a) 7871 (b) 7859 (c) 7866 (d) 7876
24. 24.Which of these numbers is divisible by 3? (a) 1092 (b) 1094 (c) 1091 (d) 1097
25. 25.Which of these numbers is divisible by 4? (a) 1916 (b) 1914 (c) 1918 (d) 1922
26. 26.What is the units digit of 2^53? (a) 8 (b) 2 (c) 4 (d) 6
27. 27.What is the remainder when 401 is divided by 7? (a) 2 (b) 5 (c) 1 (d) 3
28. 28.What is the remainder when 451 is divided by 8? (a) 3 (b) 4 (c) 2 (d) 5
29. 29.What is the smallest number you must add to 435 to make it divisible by 12? (a) 10 (b) 3 (c) 9 (d) 8
30. 30.What is the smallest number you must add to 198 to make it divisible by 4? (a) 3 (b) 1 (c) 2
31. 31.Which of these numbers is divisible by 3? (a) 2096 (b) 2098 (c) 2097 (d) 2102
32. 32.What is the remainder when 278 is divided by 11? (a) 4 (b) 2 (c) 8 (d) 3
33. 33.Which of these numbers is divisible by 3? (a) 2025 (b) 2029 (c) 2024 (d) 2027
34. 34.What is the remainder when 324 is divided by 8? (a) 5 (b) 6 (c) 3 (d) 4
35. 35.What is the units digit of 8^44? (a) 6 (b) 4 (c) 2 (d) 8
36. 36.What is the smallest number you must add to 46 to make it divisible by 4? (a) 3 (b) 1 (c) 2
37. 37.What is the remainder when 265 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
38. 38.Which of these numbers is divisible by 6? (a) 2240 (b) 2229 (c) 2232 (d) 2236
39. 39.What is the units digit of 7^21? (a) 9 (b) 3 (c) 1 (d) 7
40. 40.What is the smallest number you must add to 145 to make it divisible by 3? (a) 2 (b) 1

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