Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 741 · 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 251 is divided by 7? (a) 1 (b) 5 (c) 6 (d) 4
2. 2.What is the remainder when 191 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
3. 3.What is the smallest number you must add to 426 to make it divisible by 10? (a) 5 (b) 6 (c) 4 (d) 3
4. 4.What is the units digit of 3^34? (a) 3 (b) 7 (c) 9 (d) 1
5. 5.What is the smallest number you must add to 147 to make it divisible by 4? (a) 2 (b) 3 (c) 1
6. 6.Which of these numbers is divisible by 3? (a) 212 (b) 209 (c) 214 (d) 210
7. 7.Which of these numbers is divisible by 11? (a) 4773 (b) 4774 (c) 4780 (d) 4791
8. 8.What is the remainder when 203 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
9. 9.What is the units digit of 2^19? (a) 4 (b) 2 (c) 8 (d) 6
10. 10.What is the smallest number you must add to 115 to make it divisible by 4? (a) 3 (b) 1 (c) 2
11. 11.What is the remainder when 802 is divided by 12? (a) 11 (b) 2 (c) 9 (d) 10
12. 12.What is the units digit of 2^35? (a) 2 (b) 4 (c) 8 (d) 6
13. 13.Which of these numbers is divisible by 4? (a) 3114 (b) 3119 (c) 3116 (d) 3123
14. 14.Which of these numbers is divisible by 4? (a) 2404 (b) 2407 (c) 2409 (d) 2402
15. 15.What is the units digit of 2^38? (a) 4 (b) 2 (c) 8 (d) 6
16. 16.What is the remainder when 697 is divided by 12? (a) 0 (b) 1 (c) 2 (d) 11
17. 17.What is the smallest number you must add to 344 to make it divisible by 7? (a) 6 (b) 1 (c) 5 (d) 4
18. 18.Which of these numbers is divisible by 8? (a) 1431 (b) 1434 (c) 1443 (d) 1432
19. 19.Which of these numbers is divisible by 6? (a) 4674 (b) 4682 (c) 4679 (d) 4672
20. 20.What is the smallest number you must add to 226 to make it divisible by 3? (a) 2 (b) 1
21. 21.What is the units digit of 7^44? (a) 3 (b) 7 (c) 1 (d) 9
22. 22.What is the smallest number you must add to 170 to make it divisible by 12? (a) 11 (b) 2 (c) 10 (d) 9
23. 23.What is the smallest number you must add to 134 to make it divisible by 3? (a) 1 (b) 2
24. 24.What is the remainder when 241 is divided by 6? (a) 1 (b) 5 (c) 0 (d) 2
25. 25.Which of these numbers is divisible by 4? (a) 3405 (b) 3408 (c) 3414 (d) 3410
26. 26.What is the units digit of 7^24? (a) 7 (b) 9 (c) 3 (d) 1
27. 27.What is the units digit of 9^46? (a) 1 (b) 0 (c) 9 (d) 2
28. 28.What is the remainder when 538 is divided by 10? (a) 2 (b) 7 (c) 8 (d) 9
29. 29.What is the remainder when 318 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
30. 30.Which of these numbers is divisible by 3? (a) 292 (b) 298 (c) 294 (d) 296
31. 31.What is the smallest number you must add to 266 to make it divisible by 4? (a) 2 (b) 3 (c) 1
32. 32.What is the units digit of 9^49? (a) 8 (b) 1 (c) 7 (d) 9
33. 33.What is the smallest number you must add to 173 to make it divisible by 3? (a) 2 (b) 1
34. 34.What is the units digit of 4^40? (a) 5 (b) 7 (c) 4 (d) 6
35. 35.What is the units digit of 4^25? (a) 5 (b) 3 (c) 4 (d) 6
36. 36.What is the remainder when 262 is divided by 12? (a) 9 (b) 11 (c) 2 (d) 10
37. 37.What is the remainder when 382 is divided by 7? (a) 3 (b) 4 (c) 5 (d) 6
38. 38.Which of these numbers is divisible by 3? (a) 2318 (b) 2323 (c) 2320 (d) 2319
39. 39.What is the remainder when 154 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
40. 40.What is the remainder when 542 is divided by 7? (a) 4 (b) 5 (c) 3 (d) 2

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