Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 965 · 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 2^41? (a) 8 (b) 4 (c) 6 (d) 2
2. 2.Which of these numbers is divisible by 6? (a) 819 (b) 806 (c) 810 (d) 815
3. 3.Which of these numbers is divisible by 6? (a) 1750 (b) 1754 (c) 1752 (d) 1763
4. 4.What is the smallest number you must add to 139 to make it divisible by 9? (a) 6 (b) 5 (c) 7 (d) 4
5. 5.Which of these numbers is divisible by 8? (a) 2579 (b) 2595 (c) 2587 (d) 2584
6. 6.What is the units digit of 2^13? (a) 6 (b) 8 (c) 2 (d) 4
7. 7.What is the units digit of 3^37? (a) 3 (b) 1 (c) 9 (d) 7
8. 8.What is the units digit of 2^14? (a) 8 (b) 2 (c) 6 (d) 4
9. 9.What is the smallest number you must add to 236 to make it divisible by 9? (a) 7 (b) 8 (c) 6 (d) 2
10. 10.Which of these numbers is divisible by 6? (a) 995 (b) 990 (c) 997 (d) 988
11. 11.What is the remainder when 201 is divided by 9? (a) 4 (b) 2 (c) 3 (d) 6
12. 12.What is the remainder when 102 is divided by 8? (a) 6 (b) 2 (c) 5 (d) 7
13. 13.What is the smallest number you must add to 592 to make it divisible by 12? (a) 8 (b) 4 (c) 7 (d) 9
14. 14.What is the units digit of 8^51? (a) 8 (b) 4 (c) 6 (d) 2
15. 15.What is the remainder when 158 is divided by 3? (a) 1 (b) 0 (c) 2
16. 16.What is the smallest number you must add to 800 to make it divisible by 11? (a) 3 (b) 2 (c) 4 (d) 8
17. 17.Which of these numbers is divisible by 4? (a) 1815 (b) 1822 (c) 1816 (d) 1817
18. 18.What is the units digit of 7^21? (a) 9 (b) 1 (c) 3 (d) 7
19. 19.What is the smallest number you must add to 542 to make it divisible by 7? (a) 4 (b) 3 (c) 5 (d) 6
20. 20.What is the smallest number you must add to 262 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
21. 21.Which of these numbers is divisible by 9? (a) 3344 (b) 3335 (c) 3339 (d) 3354
22. 22.What is the remainder when 552 is divided by 7? (a) 1 (b) 4 (c) 5 (d) 6
23. 23.Which of these numbers is divisible by 6? (a) 2411 (b) 2400 (c) 2398 (d) 2405
24. 24.What is the units digit of 4^56? (a) 5 (b) 7 (c) 6 (d) 4
25. 25.What is the remainder when 116 is divided by 11? (a) 6 (b) 7 (c) 8 (d) 5
26. 26.What is the units digit of 9^40? (a) 2 (b) 0 (c) 1 (d) 9
27. 27.Which of these numbers is divisible by 8? (a) 538 (b) 533 (c) 527 (d) 528
28. 28.Which of these numbers is divisible by 11? (a) 4356 (b) 4353 (c) 4375 (d) 4361
29. 29.Which of these numbers is divisible by 6? (a) 851 (b) 837 (c) 840 (d) 845
30. 30.What is the units digit of 9^60? (a) 9 (b) 2 (c) 1 (d) 0
31. 31.What is the smallest number you must add to 427 to make it divisible by 9? (a) 6 (b) 4 (c) 7 (d) 5
32. 32.What is the units digit of 4^50? (a) 6 (b) 7 (c) 5 (d) 4
33. 33.What is the units digit of 4^27? (a) 5 (b) 3 (c) 6 (d) 4
34. 34.What is the units digit of 7^18? (a) 7 (b) 9 (c) 3 (d) 1
35. 35.What is the smallest number you must add to 185 to make it divisible by 7? (a) 5 (b) 4 (c) 6 (d) 3
36. 36.Which of these numbers is divisible by 11? (a) 5165 (b) 5170 (c) 5186 (d) 5173
37. 37.What is the remainder when 271 is divided by 6? (a) 5 (b) 2 (c) 0 (d) 1
38. 38.What is the units digit of 3^16? (a) 9 (b) 7 (c) 3 (d) 1
39. 39.What is the remainder when 68 is divided by 3? (a) 1 (b) 0 (c) 2
40. 40.What is the remainder when 114 is divided by 11? (a) 7 (b) 4 (c) 5 (d) 3

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