Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 592 · 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 remainder when 119 is divided by 6? (a) 4 (b) 5 (c) 3 (d) 1
2. 2.What is the units digit of 3^25? (a) 1 (b) 7 (c) 9 (d) 3
3. 3.Which of these numbers is divisible by 11? (a) 6502 (b) 6517 (c) 6501 (d) 6499
4. 4.What is the smallest number you must add to 341 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
5. 5.Which of these numbers is divisible by 9? (a) 2718 (b) 2729 (c) 2711 (d) 2726
6. 6.Which of these numbers is divisible by 11? (a) 6329 (b) 6305 (c) 6314 (d) 6320
7. 7.What is the remainder when 69 is divided by 5? (a) 3 (b) 4 (c) 2 (d) 1
8. 8.Which of these numbers is divisible by 11? (a) 5396 (b) 5390 (c) 5382 (d) 5409
9. 9.What is the units digit of 9^16? (a) 9 (b) 1 (c) 0 (d) 2
10. 10.Which of these numbers is divisible by 4? (a) 1225 (b) 1217 (c) 1220 (d) 1223
11. 11.What is the units digit of 4^48? (a) 4 (b) 6 (c) 7 (d) 5
12. 12.Which of these numbers is divisible by 3? (a) 748 (b) 745 (c) 747 (d) 751
13. 13.What is the units digit of 9^58? (a) 2 (b) 9 (c) 0 (d) 1
14. 14.What is the smallest number you must add to 128 to make it divisible by 3? (a) 2 (b) 1
15. 15.What is the units digit of 3^24? (a) 7 (b) 1 (c) 9 (d) 3
16. 16.What is the units digit of 8^57? (a) 6 (b) 8 (c) 2 (d) 4
17. 17.What is the remainder when 583 is divided by 12? (a) 6 (b) 5 (c) 7 (d) 8
18. 18.What is the smallest number you must add to 86 to make it divisible by 4? (a) 1 (b) 3 (c) 2
19. 19.What is the smallest number you must add to 305 to make it divisible by 8? (a) 5 (b) 1 (c) 7 (d) 6
20. 20.What is the units digit of 8^15? (a) 6 (b) 4 (c) 8 (d) 2
21. 21.What is the smallest number you must add to 662 to make it divisible by 10? (a) 2 (b) 7 (c) 8 (d) 9
22. 22.What is the remainder when 370 is divided by 8? (a) 2 (b) 1 (c) 3 (d) 6
23. 23.Which of these numbers is divisible by 3? (a) 833 (b) 839 (c) 834 (d) 835
24. 24.Which of these numbers is divisible by 8? (a) 6288 (b) 6301 (c) 6285 (d) 6294
25. 25.Which of these numbers is divisible by 3? (a) 1118 (b) 1119 (c) 1120 (d) 1124
26. 26.What is the remainder when 298 is divided by 10? (a) 8 (b) 7 (c) 2 (d) 9
27. 27.Which of these numbers is divisible by 4? (a) 2580 (b) 2587 (c) 2581 (d) 2577
28. 28.Which of these numbers is divisible by 3? (a) 2151 (b) 2153 (c) 2149 (d) 2155
29. 29.What is the smallest number you must add to 514 to make it divisible by 12? (a) 10 (b) 3 (c) 2 (d) 1
30. 30.What is the remainder when 108 is divided by 10? (a) 9 (b) 2 (c) 8 (d) 7
31. 31.What is the smallest number you must add to 226 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
32. 32.Which of these numbers is divisible by 8? (a) 768 (b) 769 (c) 780 (d) 762
33. 33.What is the remainder when 415 is divided by 8? (a) 1 (b) 6 (c) 7 (d) 5
34. 34.What is the units digit of 8^33? (a) 8 (b) 4 (c) 2 (d) 6
35. 35.What is the units digit of 2^53? (a) 8 (b) 6 (c) 4 (d) 2
36. 36.Which of these numbers is divisible by 6? (a) 1536 (b) 1543 (c) 1539 (d) 1532
37. 37.Which of these numbers is divisible by 8? (a) 7001 (b) 7000 (c) 6997 (d) 7015
38. 38.What is the remainder when 510 is divided by 12? (a) 8 (b) 7 (c) 5 (d) 6
39. 39.Which of these numbers is divisible by 8? (a) 1952 (b) 1963 (c) 1959 (d) 1945
40. 40.What is the remainder when 234 is divided by 10? (a) 5 (b) 4 (c) 6 (d) 3

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