Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 690 · 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 3? (a) 768 (b) 773 (c) 770 (d) 766
2. 2.What is the smallest number you must add to 239 to make it divisible by 10? (a) 3 (b) 2 (c) 9 (d) 1
3. 3.What is the smallest number you must add to 827 to make it divisible by 11? (a) 10 (b) 8 (c) 2 (d) 9
4. 4.What is the smallest number you must add to 231 to make it divisible by 9? (a) 6 (b) 4 (c) 3 (d) 2
5. 5.What is the smallest number you must add to 157 to make it divisible by 4? (a) 1 (b) 2 (c) 3
6. 6.What is the smallest number you must add to 296 to make it divisible by 7? (a) 5 (b) 2 (c) 4 (d) 6
7. 7.Which of these numbers is divisible by 3? (a) 1579 (b) 1574 (c) 1576 (d) 1575
8. 8.What is the smallest number you must add to 40 to make it divisible by 3? (a) 2 (b) 1
9. 9.Which of these numbers is divisible by 8? (a) 3990 (b) 3993 (c) 3979 (d) 3984
10. 10.Which of these numbers is divisible by 6? (a) 1510 (b) 1506 (c) 1516 (d) 1504
11. 11.What is the smallest number you must add to 409 to make it divisible by 11? (a) 9 (b) 10 (c) 8 (d) 2
12. 12.What is the remainder when 236 is divided by 3? (a) 2 (b) 0 (c) 1
13. 13.What is the remainder when 350 is divided by 12? (a) 2 (b) 10 (c) 3 (d) 1
14. 14.Which of these numbers is divisible by 3? (a) 1714 (b) 1713 (c) 1717 (d) 1712
15. 15.What is the units digit of 4^10? (a) 7 (b) 4 (c) 6 (d) 5
16. 16.What is the units digit of 3^55? (a) 3 (b) 7 (c) 1 (d) 9
17. 17.Which of these numbers is divisible by 3? (a) 2518 (b) 2512 (c) 2514 (d) 2515
18. 18.Which of these numbers is divisible by 8? (a) 328 (b) 327 (c) 331 (d) 342
19. 19.What is the units digit of 8^44? (a) 6 (b) 2 (c) 8 (d) 4
20. 20.What is the smallest number you must add to 122 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
21. 21.Which of these numbers is divisible by 8? (a) 4401 (b) 4397 (c) 4391 (d) 4392
22. 22.What is the units digit of 2^48? (a) 8 (b) 6 (c) 2 (d) 4
23. 23.What is the units digit of 9^56? (a) 2 (b) 0 (c) 9 (d) 1
24. 24.What is the remainder when 211 is divided by 12? (a) 6 (b) 7 (c) 8 (d) 5
25. 25.Which of these numbers is divisible by 4? (a) 2513 (b) 2517 (c) 2523 (d) 2516
26. 26.What is the units digit of 3^48? (a) 9 (b) 3 (c) 7 (d) 1
27. 27.What is the units digit of 8^48? (a) 4 (b) 2 (c) 8 (d) 6
28. 28.What is the remainder when 222 is divided by 10? (a) 1 (b) 2 (c) 3 (d) 8
29. 29.Which of these numbers is divisible by 11? (a) 4514 (b) 4499 (c) 4496 (d) 4508
30. 30.What is the smallest number you must add to 298 to make it divisible by 4? (a) 2 (b) 1 (c) 3
31. 31.What is the smallest number you must add to 143 to make it divisible by 4? (a) 3 (b) 1 (c) 2
32. 32.What is the remainder when 398 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
33. 33.What is the remainder when 70 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
34. 34.Which of these numbers is divisible by 4? (a) 484 (b) 482 (c) 487 (d) 489
35. 35.Which of these numbers is divisible by 8? (a) 2069 (b) 2063 (c) 2055 (d) 2056
36. 36.What is the smallest number you must add to 182 to make it divisible by 3? (a) 1 (b) 2
37. 37.What is the smallest number you must add to 297 to make it divisible by 10? (a) 2 (b) 7 (c) 4 (d) 3
38. 38.What is the units digit of 7^50? (a) 9 (b) 7 (c) 1 (d) 3
39. 39.Which of these numbers is divisible by 8? (a) 5650 (b) 5643 (c) 5640 (d) 5634
40. 40.What is the units digit of 3^50? (a) 9 (b) 1 (c) 7 (d) 3

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