Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 222 · 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 158 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
2. 2.What is the units digit of 8^34? (a) 8 (b) 6 (c) 2 (d) 4
3. 3.What is the units digit of 9^60? (a) 9 (b) 1 (c) 0 (d) 2
4. 4.Which of these numbers is divisible by 6? (a) 4749 (b) 4743 (c) 4753 (d) 4746
5. 5.Which of these numbers is divisible by 4? (a) 1410 (b) 1407 (c) 1401 (d) 1404
6. 6.Which of these numbers is divisible by 6? (a) 3847 (b) 3840 (c) 3836 (d) 3845
7. 7.What is the units digit of 4^18? (a) 4 (b) 6 (c) 7 (d) 5
8. 8.Which of these numbers is divisible by 3? (a) 172 (b) 169 (c) 166 (d) 168
9. 9.Which of these numbers is divisible by 6? (a) 5108 (b) 5113 (c) 5105 (d) 5106
10. 10.Which of these numbers is divisible by 6? (a) 2918 (b) 2924 (c) 2912 (d) 2916
11. 11.What is the units digit of 2^41? (a) 6 (b) 2 (c) 4 (d) 8
12. 12.What is the smallest number you must add to 411 to make it divisible by 9? (a) 2 (b) 4 (c) 3 (d) 6
13. 13.Which of these numbers is divisible by 11? (a) 2980 (b) 2991 (c) 2960 (d) 2970
14. 14.Which of these numbers is divisible by 6? (a) 1416 (b) 1415 (c) 1421 (d) 1423
15. 15.What is the smallest number you must add to 177 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
16. 16.Which of these numbers is divisible by 8? (a) 1904 (b) 1908 (c) 1916 (d) 1901
17. 17.What is the units digit of 2^52? (a) 6 (b) 8 (c) 4 (d) 2
18. 18.What is the smallest number you must add to 327 to make it divisible by 6? (a) 2 (b) 4 (c) 3 (d) 5
19. 19.What is the remainder when 160 is divided by 11? (a) 5 (b) 8 (c) 6 (d) 7
20. 20.What is the smallest number you must add to 879 to make it divisible by 11? (a) 2 (b) 1 (c) 10 (d) 3
21. 21.What is the units digit of 3^44? (a) 1 (b) 3 (c) 9 (d) 7
22. 22.What is the remainder when 148 is divided by 3? (a) 0 (b) 2 (c) 1
23. 23.What is the units digit of 9^44? (a) 2 (b) 9 (c) 0 (d) 1
24. 24.What is the units digit of 9^19? (a) 1 (b) 9 (c) 8 (d) 7
25. 25.What is the smallest number you must add to 479 to make it divisible by 7? (a) 5 (b) 3 (c) 4 (d) 6
26. 26.What is the units digit of 7^35? (a) 1 (b) 9 (c) 7 (d) 3
27. 27.Which of these numbers is divisible by 6? (a) 911 (b) 903 (c) 899 (d) 900
28. 28.What is the smallest number you must add to 197 to make it divisible by 4? (a) 1 (b) 3 (c) 2
29. 29.What is the remainder when 761 is divided by 10? (a) 2 (b) 9 (c) 1 (d) 0
30. 30.What is the smallest number you must add to 479 to make it divisible by 8? (a) 3 (b) 2 (c) 7 (d) 1
31. 31.Which of these numbers is divisible by 6? (a) 3766 (b) 3768 (c) 3777 (d) 3773
32. 32.What is the remainder when 240 is divided by 7? (a) 3 (b) 5 (c) 2 (d) 1
33. 33.What is the smallest number you must add to 786 to make it divisible by 11? (a) 7 (b) 5 (c) 6 (d) 8
34. 34.What is the remainder when 205 is divided by 9? (a) 2 (b) 6 (c) 7 (d) 8
35. 35.What is the smallest number you must add to 359 to make it divisible by 10? (a) 3 (b) 1 (c) 2 (d) 9
36. 36.What is the remainder when 314 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 1
37. 37.Which of these numbers is divisible by 3? (a) 1241 (b) 1235 (c) 1236 (d) 1238
38. 38.What is the remainder when 406 is divided by 10? (a) 7 (b) 6 (c) 4 (d) 5
39. 39.What is the smallest number you must add to 158 to make it divisible by 3? (a) 2 (b) 1
40. 40.What is the remainder when 245 is divided by 12? (a) 4 (b) 5 (c) 7 (d) 6

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