Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 404 · 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 267 is divided by 9? (a) 5 (b) 3 (c) 7 (d) 6
2. 2.What is the units digit of 2^42? (a) 2 (b) 6 (c) 8 (d) 4
3. 3.Which of these numbers is divisible by 8? (a) 4759 (b) 4752 (c) 4765 (d) 4750
4. 4.What is the smallest number you must add to 279 to make it divisible by 8? (a) 2 (b) 7 (c) 1 (d) 3
5. 5.What is the units digit of 7^49? (a) 1 (b) 7 (c) 3 (d) 9
6. 6.Which of these numbers is divisible by 9? (a) 2190 (b) 2196 (c) 2198 (d) 2208
7. 7.Which of these numbers is divisible by 3? (a) 1572 (b) 1574 (c) 1571 (d) 1577
8. 8.What is the units digit of 2^15? (a) 4 (b) 2 (c) 8 (d) 6
9. 9.What is the remainder when 115 is divided by 3? (a) 2 (b) 1 (c) 0
10. 10.What is the remainder when 151 is divided by 3? (a) 1 (b) 0 (c) 2
11. 11.What is the units digit of 4^26? (a) 4 (b) 5 (c) 6 (d) 7
12. 12.What is the remainder when 422 is divided by 7? (a) 1 (b) 3 (c) 2 (d) 5
13. 13.What is the remainder when 205 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
14. 14.Which of these numbers is divisible by 4? (a) 2639 (b) 2632 (c) 2633 (d) 2631
15. 15.Which of these numbers is divisible by 11? (a) 4147 (b) 4140 (c) 4163 (d) 4151
16. 16.What is the remainder when 285 is divided by 9? (a) 6 (b) 3 (c) 5 (d) 7
17. 17.Which of these numbers is divisible by 8? (a) 3008 (b) 3004 (c) 3021 (d) 3015
18. 18.What is the units digit of 9^58? (a) 1 (b) 0 (c) 9 (d) 2
19. 19.What is the remainder when 325 is divided by 10? (a) 7 (b) 6 (c) 5 (d) 4
20. 20.What is the smallest number you must add to 311 to make it divisible by 4? (a) 1 (b) 3 (c) 2
21. 21.What is the remainder when 137 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
22. 22.What is the units digit of 3^35? (a) 3 (b) 9 (c) 7 (d) 1
23. 23.What is the smallest number you must add to 94 to make it divisible by 3? (a) 1 (b) 2
24. 24.What is the smallest number you must add to 402 to make it divisible by 10? (a) 8 (b) 7 (c) 9 (d) 2
25. 25.Which of these numbers is divisible by 8? (a) 2601 (b) 2598 (c) 2591 (d) 2592
26. 26.Which of these numbers is divisible by 6? (a) 2777 (b) 2766 (c) 2762 (d) 2770
27. 27.Which of these numbers is divisible by 4? (a) 406 (b) 398 (c) 403 (d) 400
28. 28.What is the units digit of 8^19? (a) 4 (b) 6 (c) 8 (d) 2
29. 29.Which of these numbers is divisible by 4? (a) 1971 (b) 1973 (c) 1968 (d) 1967
30. 30.What is the units digit of 8^50? (a) 8 (b) 4 (c) 6 (d) 2
31. 31.What is the units digit of 2^47? (a) 6 (b) 8 (c) 2 (d) 4
32. 32.What is the remainder when 199 is divided by 8? (a) 7 (b) 6 (c) 5 (d) 1
33. 33.Which of these numbers is divisible by 3? (a) 1798 (b) 1804 (c) 1800 (d) 1801
34. 34.What is the units digit of 8^55? (a) 8 (b) 4 (c) 6 (d) 2
35. 35.Which of these numbers is divisible by 3? (a) 2134 (b) 2137 (c) 2133 (d) 2132
36. 36.What is the remainder when 44 is divided by 3? (a) 2 (b) 0 (c) 1
37. 37.What is the units digit of 3^52? (a) 1 (b) 9 (c) 7 (d) 3
38. 38.What is the units digit of 9^52? (a) 0 (b) 2 (c) 9 (d) 1
39. 39.What is the remainder when 211 is divided by 7? (a) 1 (b) 6 (c) 2 (d) 0
40. 40.Which of these numbers is divisible by 4? (a) 2848 (b) 2846 (c) 2855 (d) 2851

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