Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 785 · 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 41 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
2. 2.What is the units digit of 9^19? (a) 9 (b) 1 (c) 8 (d) 7
3. 3.What is the units digit of 9^11? (a) 7 (b) 9 (c) 8 (d) 1
4. 4.What is the units digit of 7^14? (a) 1 (b) 9 (c) 7 (d) 3
5. 5.Which of these numbers is divisible by 9? (a) 5660 (b) 5671 (c) 5664 (d) 5661
6. 6.What is the smallest number you must add to 429 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
7. 7.Which of these numbers is divisible by 4? (a) 3323 (b) 3325 (c) 3329 (d) 3324
8. 8.What is the units digit of 3^19? (a) 9 (b) 1 (c) 7 (d) 3
9. 9.Which of these numbers is divisible by 3? (a) 333 (b) 332 (c) 335 (d) 338
10. 10.Which of these numbers is divisible by 11? (a) 5589 (b) 5580 (c) 5603 (d) 5588
11. 11.Which of these numbers is divisible by 3? (a) 2394 (b) 2399 (c) 2395 (d) 2392
12. 12.What is the units digit of 2^17? (a) 2 (b) 4 (c) 8 (d) 6
13. 13.What is the remainder when 969 is divided by 12? (a) 3 (b) 8 (c) 10 (d) 9
14. 14.What is the units digit of 8^27? (a) 8 (b) 2 (c) 4 (d) 6
15. 15.What is the remainder when 496 is divided by 11? (a) 0 (b) 1 (c) 10 (d) 2
16. 16.What is the smallest number you must add to 323 to make it divisible by 4? (a) 3 (b) 2 (c) 1
17. 17.Which of these numbers is divisible by 8? (a) 2816 (b) 2819 (c) 2829 (d) 2812
18. 18.What is the units digit of 9^53? (a) 9 (b) 8 (c) 7 (d) 1
19. 19.What is the units digit of 4^20? (a) 4 (b) 6 (c) 7 (d) 5
20. 20.What is the smallest number you must add to 414 to make it divisible by 8? (a) 2 (b) 6 (c) 1 (d) 3
21. 21.Which of these numbers is divisible by 3? (a) 1168 (b) 1172 (c) 1167 (d) 1166
22. 22.Which of these numbers is divisible by 4? (a) 223 (b) 224 (c) 230 (d) 225
23. 23.What is the smallest number you must add to 408 to make it divisible by 7? (a) 6 (b) 4 (c) 2 (d) 5
24. 24.What is the remainder when 213 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
25. 25.What is the units digit of 3^27? (a) 1 (b) 9 (c) 7 (d) 3
26. 26.What is the remainder when 77 is divided by 3? (a) 0 (b) 1 (c) 2
27. 27.What is the remainder when 255 is divided by 8? (a) 1 (b) 7 (c) 6 (d) 5
28. 28.Which of these numbers is divisible by 3? (a) 1096 (b) 1095 (c) 1094 (d) 1099
29. 29.What is the remainder when 208 is divided by 3? (a) 1 (b) 2 (c) 0
30. 30.What is the remainder when 245 is divided by 11? (a) 4 (b) 3 (c) 2 (d) 8
31. 31.What is the remainder when 174 is divided by 7? (a) 6 (b) 1 (c) 4 (d) 5
32. 32.What is the remainder when 223 is divided by 3? (a) 1 (b) 0 (c) 2
33. 33.What is the smallest number you must add to 116 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
34. 34.What is the smallest number you must add to 775 to make it divisible by 10? (a) 5 (b) 7 (c) 6 (d) 4
35. 35.What is the smallest number you must add to 244 to make it divisible by 7? (a) 6 (b) 2 (c) 1 (d) 3
36. 36.Which of these numbers is divisible by 9? (a) 3371 (b) 3362 (c) 3357 (d) 3355
37. 37.Which of these numbers is divisible by 6? (a) 1077 (b) 1067 (c) 1073 (d) 1068
38. 38.What is the smallest number you must add to 97 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
39. 39.Which of these numbers is divisible by 3? (a) 596 (b) 590 (c) 592 (d) 591
40. 40.Which of these numbers is divisible by 4? (a) 2417 (b) 2420 (c) 2427 (d) 2421

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