Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 799 · 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 2^53? (a) 8 (b) 6 (c) 2 (d) 4
2. 2.What is the remainder when 141 is divided by 8? (a) 6 (b) 3 (c) 5 (d) 4
3. 3.What is the smallest number you must add to 289 to make it divisible by 4? (a) 3 (b) 2 (c) 1
4. 4.What is the units digit of 7^33? (a) 3 (b) 7 (c) 9 (d) 1
5. 5.What is the units digit of 8^37? (a) 4 (b) 2 (c) 6 (d) 8
6. 6.Which of these numbers is divisible by 3? (a) 169 (b) 165 (c) 164 (d) 167
7. 7.Which of these numbers is divisible by 4? (a) 1249 (b) 1245 (c) 1254 (d) 1248
8. 8.Which of these numbers is divisible by 8? (a) 4646 (b) 4661 (c) 4648 (d) 4649
9. 9.What is the remainder when 398 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
10. 10.What is the remainder when 249 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
11. 11.What is the units digit of 9^46? (a) 9 (b) 1 (c) 2 (d) 0
12. 12.What is the units digit of 7^45? (a) 3 (b) 7 (c) 9 (d) 1
13. 13.What is the units digit of 3^19? (a) 1 (b) 7 (c) 3 (d) 9
14. 14.What is the remainder when 413 is divided by 6? (a) 4 (b) 1 (c) 5 (d) 3
15. 15.What is the smallest number you must add to 101 to make it divisible by 3? (a) 2 (b) 1
16. 16.What is the smallest number you must add to 213 to make it divisible by 8? (a) 5 (b) 2 (c) 4 (d) 3
17. 17.What is the remainder when 302 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
18. 18.What is the smallest number you must add to 914 to make it divisible by 12? (a) 10 (b) 2 (c) 9 (d) 11
19. 19.What is the remainder when 659 is divided by 9? (a) 2 (b) 7 (c) 3 (d) 1
20. 20.What is the remainder when 739 is divided by 11? (a) 9 (b) 1 (c) 2 (d) 3
21. 21.What is the smallest number you must add to 103 to make it divisible by 10? (a) 7 (b) 3 (c) 6 (d) 8
22. 22.Which of these numbers is divisible by 4? (a) 1129 (b) 1132 (c) 1139 (d) 1135
23. 23.Which of these numbers is divisible by 11? (a) 7501 (b) 7483 (c) 7480 (d) 7472
24. 24.Which of these numbers is divisible by 8? (a) 4030 (b) 4024 (c) 4022 (d) 4034
25. 25.Which of these numbers is divisible by 6? (a) 531 (b) 522 (c) 520 (d) 524
26. 26.What is the units digit of 8^51? (a) 6 (b) 2 (c) 8 (d) 4
27. 27.What is the units digit of 9^16? (a) 2 (b) 0 (c) 9 (d) 1
28. 28.Which of these numbers is divisible by 8? (a) 5488 (b) 5482 (c) 5493 (d) 5500
29. 29.Which of these numbers is divisible by 3? (a) 2619 (b) 2621 (c) 2617 (d) 2623
30. 30.What is the units digit of 8^27? (a) 6 (b) 8 (c) 4 (d) 2
31. 31.Which of these numbers is divisible by 11? (a) 1865 (b) 1874 (c) 1854 (d) 1859
32. 32.Which of these numbers is divisible by 9? (a) 4755 (b) 4773 (c) 4765 (d) 4761
33. 33.What is the smallest number you must add to 265 to make it divisible by 4? (a) 1 (b) 3 (c) 2
34. 34.What is the remainder when 125 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
35. 35.What is the units digit of 2^18? (a) 6 (b) 4 (c) 2 (d) 8
36. 36.What is the remainder when 363 is divided by 8? (a) 5 (b) 3 (c) 4 (d) 2
37. 37.What is the smallest number you must add to 93 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 5
38. 38.What is the remainder when 258 is divided by 10? (a) 9 (b) 8 (c) 7 (d) 2
39. 39.What is the units digit of 3^57? (a) 1 (b) 3 (c) 9 (d) 7
40. 40.What is the units digit of 2^42? (a) 6 (b) 8 (c) 4 (d) 2

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