Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 970 · 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.Which of these numbers is divisible by 11? (a) 9009 (b) 9000 (c) 9016 (d) 9028
2. 2.Which of these numbers is divisible by 3? (a) 853 (b) 848 (c) 850 (d) 849
3. 3.What is the remainder when 210 is divided by 12? (a) 7 (b) 8 (c) 5 (d) 6
4. 4.Which of these numbers is divisible by 3? (a) 1981 (b) 1984 (c) 1980 (d) 1979
5. 5.What is the smallest number you must add to 265 to make it divisible by 9? (a) 7 (b) 5 (c) 6 (d) 4
6. 6.What is the units digit of 2^33? (a) 8 (b) 4 (c) 2 (d) 6
7. 7.Which of these numbers is divisible by 3? (a) 1409 (b) 1414 (c) 1410 (d) 1412
8. 8.What is the units digit of 8^58? (a) 8 (b) 6 (c) 4 (d) 2
9. 9.What is the remainder when 340 is divided by 8? (a) 6 (b) 4 (c) 3 (d) 5
10. 10.What is the units digit of 3^33? (a) 9 (b) 3 (c) 7 (d) 1
11. 11.Which of these numbers is divisible by 3? (a) 1170 (b) 1168 (c) 1174 (d) 1171
12. 12.What is the smallest number you must add to 118 to make it divisible by 4? (a) 3 (b) 1 (c) 2
13. 13.What is the units digit of 9^29? (a) 7 (b) 9 (c) 8 (d) 1
14. 14.Which of these numbers is divisible by 11? (a) 2928 (b) 2926 (c) 2923 (d) 2940
15. 15.What is the smallest number you must add to 253 to make it divisible by 12? (a) 11 (b) 9 (c) 10 (d) 1
16. 16.What is the remainder when 67 is divided by 3? (a) 2 (b) 0 (c) 1
17. 17.Which of these numbers is divisible by 8? (a) 1305 (b) 1323 (c) 1312 (d) 1314
18. 18.What is the units digit of 7^14? (a) 7 (b) 3 (c) 1 (d) 9
19. 19.What is the remainder when 282 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
20. 20.What is the remainder when 236 is divided by 6? (a) 3 (b) 2 (c) 1 (d) 4
21. 21.What is the remainder when 234 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
22. 22.What is the smallest number you must add to 551 to make it divisible by 10? (a) 1 (b) 8 (c) 7 (d) 9
23. 23.Which of these numbers is divisible by 8? (a) 1807 (b) 1822 (c) 1810 (d) 1808
24. 24.What is the units digit of 3^53? (a) 7 (b) 1 (c) 3 (d) 9
25. 25.What is the units digit of 2^52? (a) 2 (b) 4 (c) 6 (d) 8
26. 26.What is the units digit of 2^29? (a) 2 (b) 4 (c) 8 (d) 6
27. 27.What is the smallest number you must add to 300 to make it divisible by 8? (a) 3 (b) 5 (c) 6 (d) 4
28. 28.What is the smallest number you must add to 155 to make it divisible by 12? (a) 11 (b) 1 (c) 3 (d) 2
29. 29.What is the smallest number you must add to 35 to make it divisible by 3? (a) 1 (b) 2
30. 30.What is the smallest number you must add to 536 to make it divisible by 12? (a) 4 (b) 5 (c) 8 (d) 3
31. 31.What is the units digit of 4^33? (a) 6 (b) 4 (c) 5 (d) 3
32. 32.Which of these numbers is divisible by 3? (a) 206 (b) 207 (c) 211 (d) 209
33. 33.What is the units digit of 2^10? (a) 8 (b) 2 (c) 4 (d) 6
34. 34.Which of these numbers is divisible by 3? (a) 440 (b) 436 (c) 435 (d) 434
35. 35.What is the smallest number you must add to 416 to make it divisible by 6? (a) 4 (b) 2 (c) 5 (d) 3
36. 36.What is the smallest number you must add to 389 to make it divisible by 6? (a) 3 (b) 5 (c) 1 (d) 2
37. 37.Which of these numbers is divisible by 11? (a) 6829 (b) 6803 (c) 6810 (d) 6809
38. 38.Which of these numbers is divisible by 9? (a) 6670 (b) 6661 (c) 6669 (d) 6682
39. 39.What is the units digit of 4^38? (a) 4 (b) 6 (c) 5 (d) 7
40. 40.What is the remainder when 658 is divided by 10? (a) 7 (b) 9 (c) 8 (d) 2

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