Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 550 · 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 6? (a) 3215 (b) 3206 (c) 3217 (d) 3210
2. 2.What is the smallest number you must add to 716 to make it divisible by 9? (a) 3 (b) 5 (c) 6 (d) 4
3. 3.Which of these numbers is divisible by 6? (a) 2722 (b) 2712 (c) 2711 (d) 2716
4. 4.Which of these numbers is divisible by 3? (a) 1002 (b) 1003 (c) 1006 (d) 1000
5. 5.Which of these numbers is divisible by 6? (a) 628 (b) 631 (c) 624 (d) 622
6. 6.What is the smallest number you must add to 191 to make it divisible by 10? (a) 9 (b) 1 (c) 7 (d) 8
7. 7.What is the units digit of 9^58? (a) 0 (b) 9 (c) 2 (d) 1
8. 8.What is the smallest number you must add to 645 to make it divisible by 12? (a) 9 (b) 4 (c) 2 (d) 3
9. 9.What is the smallest number you must add to 716 to make it divisible by 11? (a) 8 (b) 9 (c) 10 (d) 1
10. 10.What is the units digit of 9^41? (a) 8 (b) 7 (c) 1 (d) 9
11. 11.What is the remainder when 599 is divided by 12? (a) 9 (b) 11 (c) 1 (d) 10
12. 12.What is the units digit of 3^39? (a) 7 (b) 3 (c) 9 (d) 1
13. 13.What is the units digit of 9^11? (a) 7 (b) 1 (c) 8 (d) 9
14. 14.What is the remainder when 533 is divided by 7? (a) 6 (b) 2 (c) 1 (d) 0
15. 15.What is the units digit of 4^50? (a) 7 (b) 4 (c) 6 (d) 5
16. 16.What is the remainder when 62 is divided by 3? (a) 0 (b) 2 (c) 1
17. 17.Which of these numbers is divisible by 8? (a) 704 (b) 719 (c) 703 (d) 705
18. 18.What is the smallest number you must add to 219 to make it divisible by 7? (a) 2 (b) 5 (c) 4 (d) 6
19. 19.What is the remainder when 463 is divided by 6? (a) 1 (b) 2 (c) 5 (d) 0
20. 20.What is the remainder when 128 is divided by 11? (a) 4 (b) 6 (c) 8 (d) 7
21. 21.What is the smallest number you must add to 58 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
22. 22.What is the remainder when 305 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
23. 23.What is the smallest number you must add to 228 to make it divisible by 11? (a) 3 (b) 4 (c) 8 (d) 2
24. 24.What is the remainder when 149 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
25. 25.What is the smallest number you must add to 53 to make it divisible by 4? (a) 3 (b) 2 (c) 1
26. 26.What is the units digit of 8^56? (a) 2 (b) 4 (c) 6 (d) 8
27. 27.What is the remainder when 727 is divided by 12? (a) 5 (b) 6 (c) 7 (d) 8
28. 28.What is the smallest number you must add to 172 to make it divisible by 3? (a) 2 (b) 1
29. 29.Which of these numbers is divisible by 3? (a) 1423 (b) 1421 (c) 1418 (d) 1419
30. 30.What is the smallest number you must add to 306 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
31. 31.What is the remainder when 682 is divided by 9? (a) 8 (b) 6 (c) 2 (d) 7
32. 32.What is the smallest number you must add to 227 to make it divisible by 10? (a) 7 (b) 2 (c) 3 (d) 4
33. 33.What is the units digit of 8^37? (a) 6 (b) 4 (c) 2 (d) 8
34. 34.Which of these numbers is divisible by 9? (a) 2235 (b) 2241 (c) 2246 (d) 2254
35. 35.What is the units digit of 2^41? (a) 8 (b) 2 (c) 4 (d) 6
36. 36.What is the units digit of 2^50? (a) 8 (b) 2 (c) 6 (d) 4
37. 37.What is the units digit of 2^56? (a) 2 (b) 8 (c) 4 (d) 6
38. 38.What is the remainder when 516 is divided by 7? (a) 5 (b) 2 (c) 4 (d) 6
39. 39.What is the remainder when 247 is divided by 6? (a) 2 (b) 5 (c) 1 (d) 0
40. 40.What is the smallest number you must add to 66 to make it divisible by 4? (a) 1 (b) 3 (c) 2

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