Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 856 · 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 units digit of 4^49? (a) 6 (b) 3 (c) 4 (d) 5
2. 2.What is the remainder when 419 is divided by 6? (a) 1 (b) 4 (c) 5 (d) 3
3. 3.What is the remainder when 769 is divided by 10? (a) 1 (b) 8 (c) 9 (d) 7
4. 4.What is the remainder when 273 is divided by 10? (a) 2 (b) 7 (c) 3 (d) 4
5. 5.What is the remainder when 322 is divided by 9? (a) 7 (b) 8 (c) 6 (d) 2
6. 6.What is the units digit of 9^27? (a) 1 (b) 7 (c) 8 (d) 9
7. 7.What is the smallest number you must add to 112 to make it divisible by 3? (a) 2 (b) 1
8. 8.What is the units digit of 2^19? (a) 6 (b) 8 (c) 4 (d) 2
9. 9.What is the smallest number you must add to 116 to make it divisible by 3? (a) 1 (b) 2
10. 10.What is the units digit of 3^41? (a) 1 (b) 3 (c) 7 (d) 9
11. 11.Which of these numbers is divisible by 4? (a) 1314 (b) 1318 (c) 1311 (d) 1312
12. 12.What is the units digit of 8^48? (a) 6 (b) 4 (c) 8 (d) 2
13. 13.What is the units digit of 2^15? (a) 6 (b) 4 (c) 2 (d) 8
14. 14.Which of these numbers is divisible by 9? (a) 4491 (b) 4505 (c) 4496 (d) 4487
15. 15.What is the units digit of 8^20? (a) 6 (b) 4 (c) 8 (d) 2
16. 16.What is the smallest number you must add to 147 to make it divisible by 10? (a) 4 (b) 7 (c) 3 (d) 2
17. 17.What is the smallest number you must add to 299 to make it divisible by 12? (a) 3 (b) 11 (c) 2 (d) 1
18. 18.What is the remainder when 229 is divided by 3? (a) 0 (b) 2 (c) 1
19. 19.Which of these numbers is divisible by 6? (a) 3954 (b) 3952 (c) 3956 (d) 3962
20. 20.What is the smallest number you must add to 430 to make it divisible by 9? (a) 1 (b) 2 (c) 3 (d) 7
21. 21.What is the smallest number you must add to 494 to make it divisible by 11? (a) 1 (b) 2 (c) 3 (d) 10
22. 22.What is the remainder when 127 is divided by 6? (a) 2 (b) 5 (c) 0 (d) 1
23. 23.What is the smallest number you must add to 211 to make it divisible by 12? (a) 5 (b) 7 (c) 4 (d) 6
24. 24.Which of these numbers is divisible by 8? (a) 426 (b) 421 (c) 424 (d) 433
25. 25.What is the smallest number you must add to 310 to make it divisible by 11? (a) 10 (b) 8 (c) 9 (d) 2
26. 26.Which of these numbers is divisible by 4? (a) 1960 (b) 1957 (c) 1963 (d) 1965
27. 27.Which of these numbers is divisible by 4? (a) 235 (b) 227 (c) 231 (d) 228
28. 28.What is the smallest number you must add to 376 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
29. 29.What is the units digit of 9^14? (a) 2 (b) 0 (c) 1 (d) 9
30. 30.Which of these numbers is divisible by 8? (a) 2773 (b) 2756 (c) 2760 (d) 2765
31. 31.What is the remainder when 288 is divided by 7? (a) 1 (b) 6 (c) 2 (d) 0
32. 32.Which of these numbers is divisible by 4? (a) 2425 (b) 2424 (c) 2431 (d) 2422
33. 33.What is the smallest number you must add to 137 to make it divisible by 8? (a) 1 (b) 7 (c) 6 (d) 5
34. 34.What is the smallest number you must add to 446 to make it divisible by 6? (a) 5 (b) 2 (c) 4 (d) 3
35. 35.Which of these numbers is divisible by 8? (a) 705 (b) 693 (c) 701 (d) 696
36. 36.What is the units digit of 2^25? (a) 2 (b) 8 (c) 6 (d) 4
37. 37.What is the units digit of 8^27? (a) 6 (b) 4 (c) 2 (d) 8
38. 38.Which of these numbers is divisible by 4? (a) 3000 (b) 3006 (c) 3002 (d) 2997
39. 39.What is the smallest number you must add to 391 to make it divisible by 6? (a) 5 (b) 3 (c) 4 (d) 1
40. 40.What is the remainder when 132 is divided by 10? (a) 3 (b) 8 (c) 1 (d) 2

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