Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 623 · 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 remainder when 498 is divided by 11? (a) 4 (b) 3 (c) 2 (d) 8
2. 2.What is the units digit of 4^20? (a) 5 (b) 7 (c) 6 (d) 4
3. 3.What is the units digit of 3^24? (a) 7 (b) 1 (c) 9 (d) 3
4. 4.What is the smallest number you must add to 174 to make it divisible by 7? (a) 2 (b) 6 (c) 1 (d) 3
5. 5.What is the smallest number you must add to 430 to make it divisible by 9? (a) 3 (b) 1 (c) 2 (d) 7
6. 6.What is the remainder when 382 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
7. 7.What is the units digit of 8^56? (a) 6 (b) 2 (c) 8 (d) 4
8. 8.Which of these numbers is divisible by 3? (a) 1977 (b) 1975 (c) 1979 (d) 1981
9. 9.What is the units digit of 2^31? (a) 4 (b) 2 (c) 8 (d) 6
10. 10.What is the remainder when 679 is divided by 10? (a) 7 (b) 8 (c) 9 (d) 1
11. 11.Which of these numbers is divisible by 9? (a) 2285 (b) 2291 (c) 2276 (d) 2277
12. 12.What is the units digit of 9^16? (a) 2 (b) 0 (c) 9 (d) 1
13. 13.Which of these numbers is divisible by 4? (a) 1778 (b) 1775 (c) 1783 (d) 1776
14. 14.Which of these numbers is divisible by 8? (a) 481 (b) 472 (c) 471 (d) 475
15. 15.What is the smallest number you must add to 178 to make it divisible by 3? (a) 1 (b) 2
16. 16.What is the units digit of 3^13? (a) 3 (b) 1 (c) 9 (d) 7
17. 17.What is the remainder when 171 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
18. 18.Which of these numbers is divisible by 11? (a) 9034 (b) 9025 (c) 9050 (d) 9031
19. 19.What is the smallest number you must add to 46 to make it divisible by 3? (a) 1 (b) 2
20. 20.What is the remainder when 337 is divided by 11? (a) 4 (b) 8 (c) 7 (d) 6
21. 21.Which of these numbers is divisible by 9? (a) 7352 (b) 7356 (c) 7353 (d) 7369
22. 22.What is the remainder when 236 is divided by 3? (a) 1 (b) 0 (c) 2
23. 23.What is the units digit of 9^13? (a) 1 (b) 8 (c) 7 (d) 9
24. 24.Which of these numbers is divisible by 8? (a) 2230 (b) 2224 (c) 2217 (d) 2238
25. 25.What is the remainder when 90 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
26. 26.What is the remainder when 70 is divided by 3? (a) 0 (b) 2 (c) 1
27. 27.What is the units digit of 3^38? (a) 7 (b) 9 (c) 1 (d) 3
28. 28.What is the units digit of 9^47? (a) 1 (b) 7 (c) 8 (d) 9
29. 29.What is the remainder when 151 is divided by 9? (a) 6 (b) 8 (c) 2 (d) 7
30. 30.What is the units digit of 2^34? (a) 4 (b) 8 (c) 2 (d) 6
31. 31.Which of these numbers is divisible by 3? (a) 1103 (b) 1101 (c) 1106 (d) 1099
32. 32.What is the remainder when 499 is divided by 7? (a) 2 (b) 5 (c) 3 (d) 1
33. 33.What is the remainder when 116 is divided by 7? (a) 4 (b) 5 (c) 3 (d) 6
34. 34.Which of these numbers is divisible by 3? (a) 984 (b) 983 (c) 988 (d) 986
35. 35.What is the units digit of 3^48? (a) 1 (b) 7 (c) 9 (d) 3
36. 36.What is the remainder when 179 is divided by 8? (a) 2 (b) 4 (c) 3 (d) 5
37. 37.What is the smallest number you must add to 158 to make it divisible by 3? (a) 1 (b) 2
38. 38.What is the smallest number you must add to 355 to make it divisible by 6? (a) 3 (b) 1 (c) 4 (d) 5
39. 39.What is the units digit of 7^54? (a) 9 (b) 3 (c) 7 (d) 1
40. 40.What is the units digit of 3^27? (a) 9 (b) 1 (c) 7 (d) 3

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