Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 543 · 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 3^21? (a) 7 (b) 9 (c) 3 (d) 1
2. 2.What is the smallest number you must add to 58 to make it divisible by 3? (a) 2 (b) 1
3. 3.What is the remainder when 218 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
4. 4.What is the remainder when 34 is divided by 3? (a) 0 (b) 2 (c) 1
5. 5.Which of these numbers is divisible by 11? (a) 7203 (b) 7210 (c) 7205 (d) 7218
6. 6.Which of these numbers is divisible by 9? (a) 5362 (b) 5372 (c) 5380 (d) 5364
7. 7.What is the smallest number you must add to 127 to make it divisible by 3? (a) 2 (b) 1
8. 8.What is the smallest number you must add to 737 to make it divisible by 12? (a) 8 (b) 6 (c) 5 (d) 7
9. 9.What is the smallest number you must add to 198 to make it divisible by 12? (a) 6 (b) 5 (c) 7 (d) 8
10. 10.What is the smallest number you must add to 111 to make it divisible by 8? (a) 2 (b) 1 (c) 7 (d) 3
11. 11.Which of these numbers is divisible by 3? (a) 300 (b) 299 (c) 302 (d) 305
12. 12.Which of these numbers is divisible by 6? (a) 1908 (b) 1919 (c) 1903 (d) 1913
13. 13.What is the units digit of 2^13? (a) 4 (b) 2 (c) 6 (d) 8
14. 14.Which of these numbers is divisible by 3? (a) 1321 (b) 1324 (c) 1327 (d) 1323
15. 15.What is the remainder when 206 is divided by 10? (a) 4 (b) 5 (c) 6 (d) 7
16. 16.Which of these numbers is divisible by 8? (a) 6074 (b) 6072 (c) 6071 (d) 6082
17. 17.Which of these numbers is divisible by 11? (a) 5648 (b) 5664 (c) 5642 (d) 5643
18. 18.What is the remainder when 51 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
19. 19.What is the remainder when 121 is divided by 3? (a) 1 (b) 2 (c) 0
20. 20.What is the smallest number you must add to 648 to make it divisible by 11? (a) 2 (b) 3 (c) 10 (d) 1
21. 21.What is the remainder when 231 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
22. 22.What is the units digit of 9^56? (a) 9 (b) 2 (c) 1 (d) 0
23. 23.Which of these numbers is divisible by 3? (a) 2143 (b) 2147 (c) 2150 (d) 2145
24. 24.What is the units digit of 2^17? (a) 4 (b) 6 (c) 8 (d) 2
25. 25.What is the units digit of 9^46? (a) 9 (b) 2 (c) 0 (d) 1
26. 26.What is the smallest number you must add to 82 to make it divisible by 3? (a) 2 (b) 1
27. 27.Which of these numbers is divisible by 11? (a) 5205 (b) 5195 (c) 5185 (d) 5192
28. 28.What is the remainder when 673 is divided by 10? (a) 2 (b) 7 (c) 3 (d) 4
29. 29.What is the units digit of 8^18? (a) 2 (b) 8 (c) 4 (d) 6
30. 30.What is the smallest number you must add to 430 to make it divisible by 6? (a) 2 (b) 3 (c) 1 (d) 4
31. 31.Which of these numbers is divisible by 4? (a) 2626 (b) 2634 (c) 2628 (d) 2630
32. 32.What is the remainder when 213 is divided by 10? (a) 7 (b) 4 (c) 3 (d) 2
33. 33.What is the units digit of 9^53? (a) 1 (b) 8 (c) 7 (d) 9
34. 34.What is the units digit of 7^46? (a) 1 (b) 9 (c) 3 (d) 7
35. 35.What is the smallest number you must add to 525 to make it divisible by 10? (a) 4 (b) 7 (c) 5 (d) 6
36. 36.Which of these numbers is divisible by 3? (a) 331 (b) 329 (c) 326 (d) 327
37. 37.What is the smallest number you must add to 73 to make it divisible by 3? (a) 1 (b) 2
38. 38.What is the smallest number you must add to 332 to make it divisible by 10? (a) 9 (b) 7 (c) 2 (d) 8
39. 39.What is the smallest number you must add to 425 to make it divisible by 11? (a) 5 (b) 7 (c) 3 (d) 4
40. 40.What is the smallest number you must add to 445 to make it divisible by 10? (a) 4 (b) 5 (c) 7 (d) 6

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