Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 516 · 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 7^11? (a) 9 (b) 1 (c) 7 (d) 3
2. 2.Which of these numbers is divisible by 3? (a) 632 (b) 634 (c) 628 (d) 630
3. 3.What is the remainder when 243 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
4. 4.Which of these numbers is divisible by 8? (a) 5669 (b) 5662 (c) 5675 (d) 5664
5. 5.Which of these numbers is divisible by 3? (a) 2193 (b) 2194 (c) 2198 (d) 2191
6. 6.What is the smallest number you must add to 549 to make it divisible by 12? (a) 4 (b) 2 (c) 3 (d) 9
7. 7.What is the remainder when 393 is divided by 11? (a) 9 (b) 8 (c) 7 (d) 3
8. 8.Which of these numbers is divisible by 3? (a) 2018 (b) 2014 (c) 2011 (d) 2013
9. 9.Which of these numbers is divisible by 6? (a) 5385 (b) 5375 (c) 5376 (d) 5380
10. 10.What is the remainder when 306 is divided by 5? (a) 1 (b) 2 (c) 0 (d) 4
11. 11.Which of these numbers is divisible by 11? (a) 9382 (b) 9388 (c) 9401 (d) 9383
12. 12.Which of these numbers is divisible by 4? (a) 1990 (b) 1993 (c) 1997 (d) 1992
13. 13.What is the units digit of 2^39? (a) 4 (b) 6 (c) 2 (d) 8
14. 14.What is the smallest number you must add to 547 to make it divisible by 8? (a) 6 (b) 5 (c) 4 (d) 3
15. 15.What is the smallest number you must add to 627 to make it divisible by 9? (a) 2 (b) 4 (c) 6 (d) 3
16. 16.What is the remainder when 547 is divided by 7? (a) 6 (b) 2 (c) 0 (d) 1
17. 17.What is the remainder when 177 is divided by 12? (a) 10 (b) 9 (c) 3 (d) 8
18. 18.Which of these numbers is divisible by 3? (a) 2194 (b) 2190 (c) 2189 (d) 2192
19. 19.Which of these numbers is divisible by 6? (a) 3986 (b) 3990 (c) 4001 (d) 3995
20. 20.What is the units digit of 7^57? (a) 3 (b) 7 (c) 9 (d) 1
21. 21.Which of these numbers is divisible by 6? (a) 5034 (b) 5038 (c) 5042 (d) 5033
22. 22.Which of these numbers is divisible by 6? (a) 3137 (b) 3122 (c) 3126 (d) 3127
23. 23.What is the remainder when 529 is divided by 10? (a) 9 (b) 8 (c) 7 (d) 1
24. 24.What is the smallest number you must add to 398 to make it divisible by 9? (a) 2 (b) 7 (c) 6 (d) 8
25. 25.What is the units digit of 8^27? (a) 8 (b) 6 (c) 4 (d) 2
26. 26.What is the units digit of 8^49? (a) 4 (b) 8 (c) 6 (d) 2
27. 27.What is the remainder when 86 is divided by 7? (a) 3 (b) 2 (c) 5 (d) 1
28. 28.What is the units digit of 3^23? (a) 3 (b) 1 (c) 7 (d) 9
29. 29.What is the units digit of 8^29? (a) 4 (b) 2 (c) 8 (d) 6
30. 30.What is the units digit of 7^23? (a) 3 (b) 1 (c) 9 (d) 7
31. 31.Which of these numbers is divisible by 4? (a) 2773 (b) 2770 (c) 2772 (d) 2779
32. 32.Which of these numbers is divisible by 6? (a) 5331 (b) 5328 (c) 5336 (d) 5325
33. 33.Which of these numbers is divisible by 4? (a) 2147 (b) 2141 (c) 2144 (d) 2149
34. 34.What is the remainder when 223 is divided by 3? (a) 1 (b) 2 (c) 0
35. 35.What is the smallest number you must add to 53 to make it divisible by 3? (a) 2 (b) 1
36. 36.What is the smallest number you must add to 242 to make it divisible by 4? (a) 1 (b) 3 (c) 2
37. 37.What is the units digit of 2^51? (a) 4 (b) 2 (c) 6 (d) 8
38. 38.What is the smallest number you must add to 527 to make it divisible by 9? (a) 5 (b) 3 (c) 4 (d) 6
39. 39.What is the remainder when 153 is divided by 6? (a) 5 (b) 4 (c) 2 (d) 3
40. 40.What is the remainder when 233 is divided by 10? (a) 2 (b) 7 (c) 4 (d) 3

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