Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 515 · 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 2^39? (a) 6 (b) 4 (c) 2 (d) 8
2. 2.What is the smallest number you must add to 113 to make it divisible by 10? (a) 8 (b) 6 (c) 7 (d) 3
3. 3.What is the smallest number you must add to 287 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
4. 4.What is the units digit of 9^13? (a) 9 (b) 1 (c) 7 (d) 8
5. 5.What is the units digit of 2^19? (a) 4 (b) 6 (c) 8 (d) 2
6. 6.What is the units digit of 3^32? (a) 9 (b) 7 (c) 3 (d) 1
7. 7.What is the smallest number you must add to 601 to make it divisible by 9? (a) 1 (b) 3 (c) 7 (d) 2
8. 8.Which of these numbers is divisible by 11? (a) 3522 (b) 3507 (c) 3519 (d) 3509
9. 9.What is the units digit of 3^56? (a) 7 (b) 3 (c) 1 (d) 9
10. 10.What is the remainder when 204 is divided by 9? (a) 5 (b) 3 (c) 7 (d) 6
11. 11.What is the units digit of 2^14? (a) 8 (b) 4 (c) 2 (d) 6
12. 12.What is the smallest number you must add to 177 to make it divisible by 11? (a) 1 (b) 10 (c) 8 (d) 9
13. 13.Which of these numbers is divisible by 4? (a) 797 (b) 790 (c) 792 (d) 795
14. 14.What is the units digit of 4^50? (a) 7 (b) 6 (c) 5 (d) 4
15. 15.Which of these numbers is divisible by 11? (a) 7333 (b) 7351 (c) 7337 (d) 7340
16. 16.What is the remainder when 434 is divided by 8? (a) 2 (b) 3 (c) 1 (d) 6
17. 17.What is the smallest number you must add to 711 to make it divisible by 11? (a) 4 (b) 7 (c) 3 (d) 5
18. 18.Which of these numbers is divisible by 3? (a) 1006 (b) 1005 (c) 1009 (d) 1003
19. 19.Which of these numbers is divisible by 6? (a) 4893 (b) 4899 (c) 4904 (d) 4896
20. 20.What is the remainder when 252 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
21. 21.What is the smallest number you must add to 218 to make it divisible by 7? (a) 6 (b) 4 (c) 1 (d) 5
22. 22.What is the smallest number you must add to 809 to make it divisible by 12? (a) 6 (b) 8 (c) 7 (d) 5
23. 23.What is the units digit of 2^52? (a) 6 (b) 8 (c) 2 (d) 4
24. 24.What is the units digit of 8^42? (a) 2 (b) 6 (c) 4 (d) 8
25. 25.Which of these numbers is divisible by 4? (a) 1910 (b) 1913 (c) 1912 (d) 1918
26. 26.What is the units digit of 3^37? (a) 7 (b) 3 (c) 9 (d) 1
27. 27.What is the smallest number you must add to 329 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
28. 28.What is the remainder when 211 is divided by 3? (a) 2 (b) 0 (c) 1
29. 29.Which of these numbers is divisible by 3? (a) 1465 (b) 1469 (c) 1464 (d) 1463
30. 30.What is the units digit of 4^24? (a) 4 (b) 6 (c) 5 (d) 7
31. 31.What is the units digit of 2^51? (a) 8 (b) 4 (c) 6 (d) 2
32. 32.Which of these numbers is divisible by 8? (a) 1223 (b) 1225 (c) 1210 (d) 1216
33. 33.What is the remainder when 66 is divided by 5? (a) 1 (b) 2 (c) 0 (d) 4
34. 34.What is the smallest number you must add to 209 to make it divisible by 3? (a) 1 (b) 2
35. 35.What is the smallest number you must add to 82 to make it divisible by 3? (a) 2 (b) 1
36. 36.What is the remainder when 273 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
37. 37.Which of these numbers is divisible by 4? (a) 179 (b) 183 (c) 186 (d) 180
38. 38.Which of these numbers is divisible by 11? (a) 1256 (b) 1249 (c) 1254 (d) 1272
39. 39.What is the units digit of 2^27? (a) 6 (b) 2 (c) 8 (d) 4
40. 40.What is the remainder when 352 is divided by 6? (a) 4 (b) 3 (c) 5 (d) 2

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