Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 863 · 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 9^29? (a) 7 (b) 9 (c) 1 (d) 8
2. 2.What is the remainder when 711 is divided by 10? (a) 1 (b) 0 (c) 2 (d) 9
3. 3.What is the units digit of 3^38? (a) 1 (b) 7 (c) 3 (d) 9
4. 4.What is the units digit of 4^16? (a) 4 (b) 6 (c) 5 (d) 7
5. 5.What is the smallest number you must add to 231 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
6. 6.What is the smallest number you must add to 611 to make it divisible by 10? (a) 7 (b) 9 (c) 8 (d) 1
7. 7.Which of these numbers is divisible by 9? (a) 6508 (b) 6517 (c) 6527 (d) 6516
8. 8.What is the smallest number you must add to 591 to make it divisible by 11? (a) 4 (b) 2 (c) 8 (d) 3
9. 9.What is the remainder when 784 is divided by 11? (a) 3 (b) 4 (c) 8 (d) 2
10. 10.What is the units digit of 8^13? (a) 4 (b) 6 (c) 2 (d) 8
11. 11.What is the units digit of 2^22? (a) 8 (b) 4 (c) 2 (d) 6
12. 12.What is the units digit of 2^21? (a) 4 (b) 6 (c) 2 (d) 8
13. 13.What is the remainder when 391 is divided by 7? (a) 6 (b) 5 (c) 1 (d) 4
14. 14.What is the smallest number you must add to 251 to make it divisible by 4? (a) 3 (b) 1 (c) 2
15. 15.What is the units digit of 2^57? (a) 2 (b) 6 (c) 8 (d) 4
16. 16.Which of these numbers is divisible by 3? (a) 2195 (b) 2196 (c) 2200 (d) 2197
17. 17.Which of these numbers is divisible by 6? (a) 1488 (b) 1496 (c) 1483 (d) 1492
18. 18.Which of these numbers is divisible by 8? (a) 4181 (b) 4168 (c) 4165 (d) 4169
19. 19.Which of these numbers is divisible by 9? (a) 5193 (b) 5196 (c) 5190 (d) 5207
20. 20.What is the smallest number you must add to 340 to make it divisible by 7? (a) 4 (b) 2 (c) 5 (d) 3
21. 21.What is the smallest number you must add to 372 to make it divisible by 11? (a) 1 (b) 3 (c) 2 (d) 9
22. 22.Which of these numbers is divisible by 3? (a) 1309 (b) 1315 (c) 1312 (d) 1311
23. 23.What is the units digit of 8^31? (a) 8 (b) 6 (c) 2 (d) 4
24. 24.What is the units digit of 3^47? (a) 9 (b) 1 (c) 3 (d) 7
25. 25.What is the remainder when 208 is divided by 3? (a) 0 (b) 1 (c) 2
26. 26.What is the units digit of 8^58? (a) 6 (b) 4 (c) 2 (d) 8
27. 27.What is the units digit of 8^10? (a) 8 (b) 4 (c) 6 (d) 2
28. 28.What is the smallest number you must add to 190 to make it divisible by 9? (a) 1 (b) 6 (c) 7 (d) 8
29. 29.What is the remainder when 890 is divided by 12? (a) 2 (b) 3 (c) 1 (d) 10
30. 30.Which of these numbers is divisible by 8? (a) 4626 (b) 4624 (c) 4635 (d) 4620
31. 31.Which of these numbers is divisible by 11? (a) 7411 (b) 7414 (c) 7434 (d) 7423
32. 32.What is the remainder when 791 is divided by 11? (a) 9 (b) 8 (c) 1 (d) 10
33. 33.Which of these numbers is divisible by 3? (a) 1021 (b) 1019 (c) 1015 (d) 1017
34. 34.What is the remainder when 140 is divided by 3? (a) 0 (b) 1 (c) 2
35. 35.Which of these numbers is divisible by 3? (a) 1462 (b) 1458 (c) 1456 (d) 1459
36. 36.Which of these numbers is divisible by 4? (a) 572 (b) 571 (c) 577 (d) 574
37. 37.What is the units digit of 3^41? (a) 9 (b) 7 (c) 1 (d) 3
38. 38.Which of these numbers is divisible by 11? (a) 8395 (b) 8411 (c) 8384 (d) 8393
39. 39.What is the smallest number you must add to 543 to make it divisible by 12? (a) 9 (b) 8 (c) 3 (d) 10
40. 40.What is the units digit of 4^42? (a) 6 (b) 4 (c) 5 (d) 7

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