Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 236 · 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 11? (a) 1119 (b) 1139 (c) 1122 (d) 1129
2. 2.What is the units digit of 4^52? (a) 5 (b) 7 (c) 6 (d) 4
3. 3.What is the smallest number you must add to 89 to make it divisible by 8? (a) 5 (b) 7 (c) 1 (d) 6
4. 4.What is the smallest number you must add to 378 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
5. 5.Which of these numbers is divisible by 6? (a) 429 (b) 435 (c) 432 (d) 439
6. 6.Which of these numbers is divisible by 4? (a) 1889 (b) 1884 (c) 1886 (d) 1882
7. 7.What is the units digit of 8^34? (a) 6 (b) 2 (c) 8 (d) 4
8. 8.What is the remainder when 773 is divided by 11? (a) 3 (b) 8 (c) 2 (d) 4
9. 9.What is the smallest number you must add to 233 to make it divisible by 4? (a) 3 (b) 2 (c) 1
10. 10.Which of these numbers is divisible by 9? (a) 2485 (b) 2469 (c) 2476 (d) 2475
11. 11.Which of these numbers is divisible by 4? (a) 607 (b) 604 (c) 609 (d) 602
12. 12.What is the units digit of 2^26? (a) 2 (b) 6 (c) 4 (d) 8
13. 13.What is the remainder when 124 is divided by 5? (a) 4 (b) 3 (c) 1 (d) 2
14. 14.What is the units digit of 8^25? (a) 6 (b) 2 (c) 4 (d) 8
15. 15.What is the remainder when 392 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
16. 16.What is the remainder when 580 is divided by 9? (a) 5 (b) 6 (c) 3 (d) 4
17. 17.What is the smallest number you must add to 156 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
18. 18.What is the units digit of 7^39? (a) 3 (b) 9 (c) 7 (d) 1
19. 19.What is the remainder when 256 is divided by 6? (a) 5 (b) 3 (c) 4 (d) 2
20. 20.What is the remainder when 222 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
21. 21.What is the smallest number you must add to 304 to make it divisible by 6? (a) 1 (b) 4 (c) 2 (d) 3
22. 22.Which of these numbers is divisible by 6? (a) 353 (b) 337 (c) 342 (d) 346
23. 23.What is the smallest number you must add to 225 to make it divisible by 4? (a) 2 (b) 1 (c) 3
24. 24.What is the remainder when 133 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
25. 25.What is the remainder when 712 is divided by 12? (a) 8 (b) 4 (c) 3 (d) 5
26. 26.What is the units digit of 7^54? (a) 1 (b) 9 (c) 3 (d) 7
27. 27.What is the units digit of 7^58? (a) 3 (b) 1 (c) 7 (d) 9
28. 28.What is the units digit of 9^13? (a) 1 (b) 9 (c) 7 (d) 8
29. 29.What is the smallest number you must add to 433 to make it divisible by 6? (a) 1 (b) 3 (c) 5 (d) 4
30. 30.What is the smallest number you must add to 61 to make it divisible by 4? (a) 1 (b) 3 (c) 2
31. 31.What is the smallest number you must add to 163 to make it divisible by 6? (a) 1 (b) 4 (c) 3 (d) 5
32. 32.What is the smallest number you must add to 267 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
33. 33.Which of these numbers is divisible by 3? (a) 2241 (b) 2243 (c) 2245 (d) 2240
34. 34.What is the smallest number you must add to 714 to make it divisible by 9? (a) 5 (b) 6 (c) 3 (d) 7
35. 35.Which of these numbers is divisible by 4? (a) 3512 (b) 3513 (c) 3511 (d) 3517
36. 36.Which of these numbers is divisible by 3? (a) 2335 (b) 2329 (c) 2332 (d) 2331
37. 37.Which of these numbers is divisible by 9? (a) 6540 (b) 6531 (c) 6547 (d) 6534
38. 38.What is the remainder when 148 is divided by 3? (a) 0 (b) 1 (c) 2
39. 39.Which of these numbers is divisible by 4? (a) 2796 (b) 2803 (c) 2794 (d) 2799
40. 40.What is the remainder when 579 is divided by 11? (a) 7 (b) 4 (c) 6 (d) 8

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