Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 372 · 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.Which of these numbers is divisible by 4? (a) 632 (b) 631 (c) 635 (d) 639
2. 2.What is the smallest number you must add to 173 to make it divisible by 11? (a) 8 (b) 4 (c) 2 (d) 3
3. 3.Which of these numbers is divisible by 3? (a) 1434 (b) 1435 (c) 1439 (d) 1433
4. 4.Which of these numbers is divisible by 6? (a) 528 (b) 530 (c) 537 (d) 524
5. 5.Which of these numbers is divisible by 4? (a) 280 (b) 287 (c) 278 (d) 282
6. 6.Which of these numbers is divisible by 4? (a) 979 (b) 975 (c) 981 (d) 976
7. 7.Which of these numbers is divisible by 3? (a) 1718 (b) 1719 (c) 1720 (d) 1724
8. 8.What is the remainder when 927 is divided by 12? (a) 3 (b) 2 (c) 9 (d) 4
9. 9.Which of these numbers is divisible by 9? (a) 949 (b) 936 (c) 933 (d) 940
10. 10.What is the smallest number you must add to 148 to make it divisible by 7? (a) 5 (b) 4 (c) 1 (d) 6
11. 11.Which of these numbers is divisible by 6? (a) 3795 (b) 3800 (c) 3806 (d) 3798
12. 12.What is the remainder when 265 is divided by 6? (a) 5 (b) 0 (c) 2 (d) 1
13. 13.What is the remainder when 134 is divided by 3? (a) 1 (b) 2 (c) 0
14. 14.Which of these numbers is divisible by 6? (a) 525 (b) 522 (c) 530 (d) 521
15. 15.What is the units digit of 2^53? (a) 8 (b) 2 (c) 4 (d) 6
16. 16.Which of these numbers is divisible by 3? (a) 390 (b) 389 (c) 391 (d) 395
17. 17.What is the units digit of 8^22? (a) 8 (b) 2 (c) 6 (d) 4
18. 18.What is the units digit of 8^13? (a) 2 (b) 8 (c) 6 (d) 4
19. 19.What is the remainder when 467 is divided by 9? (a) 8 (b) 7 (c) 6 (d) 1
20. 20.Which of these numbers is divisible by 4? (a) 2930 (b) 2928 (c) 2926 (d) 2933
21. 21.What is the remainder when 177 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
22. 22.Which of these numbers is divisible by 3? (a) 1957 (b) 1952 (c) 1953 (d) 1954
23. 23.What is the smallest number you must add to 156 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 2
24. 24.What is the remainder when 460 is divided by 6? (a) 5 (b) 2 (c) 4 (d) 3
25. 25.What is the smallest number you must add to 169 to make it divisible by 3? (a) 1 (b) 2
26. 26.What is the remainder when 522 is divided by 10? (a) 3 (b) 8 (c) 1 (d) 2
27. 27.What is the units digit of 3^59? (a) 3 (b) 7 (c) 9 (d) 1
28. 28.What is the remainder when 333 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
29. 29.Which of these numbers is divisible by 3? (a) 1864 (b) 1867 (c) 1866 (d) 1871
30. 30.What is the units digit of 9^30? (a) 9 (b) 1 (c) 0 (d) 2
31. 31.Which of these numbers is divisible by 8? (a) 7084 (b) 7073 (c) 7072 (d) 7070
32. 32.Which of these numbers is divisible by 6? (a) 1289 (b) 1292 (c) 1283 (d) 1284
33. 33.What is the units digit of 2^43? (a) 4 (b) 6 (c) 2 (d) 8
34. 34.What is the remainder when 484 is divided by 8? (a) 3 (b) 5 (c) 6 (d) 4
35. 35.What is the remainder when 205 is divided by 12? (a) 0 (b) 11 (c) 1 (d) 2
36. 36.What is the smallest number you must add to 102 to make it divisible by 9? (a) 3 (b) 6 (c) 5 (d) 7
37. 37.What is the remainder when 401 is divided by 10? (a) 0 (b) 1 (c) 9 (d) 2
38. 38.What is the remainder when 166 is divided by 3? (a) 2 (b) 0 (c) 1
39. 39.What is the units digit of 8^59? (a) 8 (b) 4 (c) 2 (d) 6
40. 40.What is the remainder when 315 is divided by 6? (a) 3 (b) 2 (c) 4 (d) 5

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