Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 982 · 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 2^36? (a) 8 (b) 2 (c) 6 (d) 4
2. 2.Which of these numbers is divisible by 6? (a) 908 (b) 902 (c) 913 (d) 906
3. 3.What is the smallest number you must add to 719 to make it divisible by 11? (a) 4 (b) 8 (c) 6 (d) 7
4. 4.Which of these numbers is divisible by 8? (a) 1577 (b) 1589 (c) 1599 (d) 1584
5. 5.What is the units digit of 2^23? (a) 4 (b) 6 (c) 2 (d) 8
6. 6.What is the remainder when 513 is divided by 12? (a) 8 (b) 3 (c) 10 (d) 9
7. 7.What is the units digit of 9^49? (a) 8 (b) 7 (c) 9 (d) 1
8. 8.What is the units digit of 8^60? (a) 4 (b) 2 (c) 6 (d) 8
9. 9.What is the smallest number you must add to 117 to make it divisible by 4? (a) 3 (b) 1 (c) 2
10. 10.What is the units digit of 7^37? (a) 9 (b) 3 (c) 7 (d) 1
11. 11.What is the units digit of 3^46? (a) 9 (b) 3 (c) 7 (d) 1
12. 12.Which of these numbers is divisible by 6? (a) 3979 (b) 3978 (c) 3987 (d) 3975
13. 13.What is the smallest number you must add to 139 to make it divisible by 4? (a) 3 (b) 2 (c) 1
14. 14.Which of these numbers is divisible by 11? (a) 8039 (b) 8049 (c) 8029 (d) 8030
15. 15.What is the remainder when 778 is divided by 11? (a) 9 (b) 7 (c) 8 (d) 3
16. 16.What is the smallest number you must add to 128 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 4
17. 17.What is the smallest number you must add to 822 to make it divisible by 12? (a) 8 (b) 5 (c) 7 (d) 6
18. 18.What is the smallest number you must add to 176 to make it divisible by 3? (a) 1 (b) 2
19. 19.Which of these numbers is divisible by 3? (a) 207 (b) 209 (c) 211 (d) 205
20. 20.Which of these numbers is divisible by 4? (a) 3422 (b) 3427 (c) 3430 (d) 3424
21. 21.What is the smallest number you must add to 193 to make it divisible by 12? (a) 9 (b) 10 (c) 1 (d) 11
22. 22.What is the smallest number you must add to 514 to make it divisible by 9? (a) 6 (b) 8 (c) 1 (d) 7
23. 23.What is the smallest number you must add to 120 to make it divisible by 7? (a) 1 (b) 4 (c) 6 (d) 5
24. 24.Which of these numbers is divisible by 3? (a) 2180 (b) 2176 (c) 2174 (d) 2175
25. 25.What is the smallest number you must add to 290 to make it divisible by 4? (a) 1 (b) 3 (c) 2
26. 26.What is the units digit of 4^40? (a) 6 (b) 7 (c) 5 (d) 4
27. 27.Which of these numbers is divisible by 6? (a) 4486 (b) 4474 (c) 4476 (d) 4480
28. 28.Which of these numbers is divisible by 3? (a) 878 (b) 880 (c) 879 (d) 884
29. 29.Which of these numbers is divisible by 3? (a) 1894 (b) 1901 (c) 1897 (d) 1896
30. 30.What is the remainder when 105 is divided by 10? (a) 7 (b) 4 (c) 6 (d) 5
31. 31.What is the remainder when 333 is divided by 6? (a) 4 (b) 5 (c) 2 (d) 3
32. 32.Which of these numbers is divisible by 6? (a) 5065 (b) 5064 (c) 5060 (d) 5075
33. 33.What is the smallest number you must add to 133 to make it divisible by 4? (a) 2 (b) 1 (c) 3
34. 34.What is the units digit of 3^50? (a) 1 (b) 9 (c) 3 (d) 7
35. 35.What is the remainder when 235 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
36. 36.Which of these numbers is divisible by 9? (a) 7383 (b) 7403 (c) 7389 (d) 7395
37. 37.What is the units digit of 8^33? (a) 4 (b) 2 (c) 8 (d) 6
38. 38.What is the units digit of 4^45? (a) 6 (b) 5 (c) 4 (d) 3
39. 39.What is the units digit of 7^51? (a) 9 (b) 7 (c) 3 (d) 1
40. 40.What is the remainder when 43 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad