Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 606 · 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 smallest number you must add to 563 to make it divisible by 12? (a) 3 (b) 2 (c) 11 (d) 1
2. 2.Which of these numbers is divisible by 8? (a) 5645 (b) 5627 (c) 5635 (d) 5632
3. 3.What is the remainder when 228 is divided by 7? (a) 6 (b) 5 (c) 3 (d) 4
4. 4.What is the units digit of 4^19? (a) 6 (b) 5 (c) 4 (d) 3
5. 5.What is the units digit of 3^16? (a) 3 (b) 1 (c) 7 (d) 9
6. 6.What is the units digit of 8^56? (a) 4 (b) 8 (c) 2 (d) 6
7. 7.What is the units digit of 3^17? (a) 7 (b) 9 (c) 1 (d) 3
8. 8.Which of these numbers is divisible by 4? (a) 1883 (b) 1879 (c) 1876 (d) 1874
9. 9.Which of these numbers is divisible by 9? (a) 8074 (b) 8082 (c) 8083 (d) 8097
10. 10.What is the remainder when 275 is divided by 7? (a) 1 (b) 5 (c) 3 (d) 2
11. 11.What is the smallest number you must add to 353 to make it divisible by 8? (a) 5 (b) 1 (c) 7 (d) 6
12. 12.What is the remainder when 380 is divided by 7? (a) 1 (b) 2 (c) 3 (d) 5
13. 13.Which of these numbers is divisible by 11? (a) 786 (b) 775 (c) 761 (d) 770
14. 14.Which of these numbers is divisible by 3? (a) 2344 (b) 2340 (c) 2341 (d) 2338
15. 15.Which of these numbers is divisible by 11? (a) 1100 (b) 1095 (c) 1103 (d) 1113
16. 16.What is the units digit of 7^60? (a) 3 (b) 7 (c) 9 (d) 1
17. 17.What is the remainder when 99 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
18. 18.Which of these numbers is divisible by 3? (a) 1019 (b) 1024 (c) 1021 (d) 1020
19. 19.Which of these numbers is divisible by 9? (a) 5022 (b) 5035 (c) 5027 (d) 5021
20. 20.Which of these numbers is divisible by 4? (a) 2077 (b) 2083 (c) 2075 (d) 2076
21. 21.What is the remainder when 123 is divided by 10? (a) 2 (b) 7 (c) 4 (d) 3
22. 22.Which of these numbers is divisible by 4? (a) 3509 (b) 3504 (c) 3506 (d) 3503
23. 23.What is the smallest number you must add to 157 to make it divisible by 4? (a) 3 (b) 2 (c) 1
24. 24.What is the remainder when 101 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 0
25. 25.What is the remainder when 98 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
26. 26.What is the units digit of 3^57? (a) 9 (b) 3 (c) 1 (d) 7
27. 27.What is the remainder when 194 is divided by 3? (a) 2 (b) 1 (c) 0
28. 28.What is the remainder when 308 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
29. 29.What is the units digit of 4^47? (a) 5 (b) 4 (c) 3 (d) 6
30. 30.What is the smallest number you must add to 850 to make it divisible by 11? (a) 3 (b) 9 (c) 8 (d) 7
31. 31.Which of these numbers is divisible by 11? (a) 8151 (b) 8155 (c) 8146 (d) 8165
32. 32.What is the remainder when 148 is divided by 3? (a) 1 (b) 0 (c) 2
33. 33.What is the units digit of 9^60? (a) 2 (b) 9 (c) 0 (d) 1
34. 34.What is the units digit of 9^42? (a) 0 (b) 1 (c) 2 (d) 9
35. 35.What is the smallest number you must add to 564 to make it divisible by 8? (a) 4 (b) 6 (c) 5 (d) 3
36. 36.What is the remainder when 733 is divided by 10? (a) 2 (b) 4 (c) 3 (d) 7
37. 37.What is the remainder when 90 is divided by 8? (a) 6 (b) 3 (c) 1 (d) 2
38. 38.Which of these numbers is divisible by 3? (a) 172 (b) 178 (c) 174 (d) 176
39. 39.Which of these numbers is divisible by 8? (a) 5794 (b) 5809 (c) 5803 (d) 5800
40. 40.Which of these numbers is divisible by 11? (a) 1848 (b) 1841 (c) 1850 (d) 1866

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