Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 634 · 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 9^41? (a) 7 (b) 8 (c) 9 (d) 1
2. 2.What is the units digit of 3^55? (a) 9 (b) 1 (c) 7 (d) 3
3. 3.What is the units digit of 8^33? (a) 2 (b) 4 (c) 6 (d) 8
4. 4.What is the remainder when 501 is divided by 9? (a) 3 (b) 7 (c) 6 (d) 5
5. 5.What is the units digit of 2^17? (a) 6 (b) 8 (c) 4 (d) 2
6. 6.What is the smallest number you must add to 315 to make it divisible by 4? (a) 3 (b) 2 (c) 1
7. 7.What is the units digit of 2^57? (a) 6 (b) 4 (c) 8 (d) 2
8. 8.What is the smallest number you must add to 433 to make it divisible by 7? (a) 2 (b) 1 (c) 6 (d) 3
9. 9.What is the units digit of 2^28? (a) 8 (b) 2 (c) 4 (d) 6
10. 10.Which of these numbers is divisible by 11? (a) 5542 (b) 5533 (c) 5551 (d) 5530
11. 11.What is the smallest number you must add to 94 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
12. 12.What is the smallest number you must add to 155 to make it divisible by 4? (a) 1 (b) 3 (c) 2
13. 13.Which of these numbers is divisible by 9? (a) 7295 (b) 7300 (c) 7310 (d) 7299
14. 14.What is the units digit of 7^52? (a) 3 (b) 7 (c) 9 (d) 1
15. 15.Which of these numbers is divisible by 9? (a) 7947 (b) 7951 (c) 7945 (d) 7960
16. 16.What is the remainder when 229 is divided by 3? (a) 2 (b) 1 (c) 0
17. 17.What is the units digit of 8^49? (a) 4 (b) 8 (c) 2 (d) 6
18. 18.Which of these numbers is divisible by 9? (a) 6364 (b) 6360 (c) 6363 (d) 6374
19. 19.Which of these numbers is divisible by 9? (a) 3843 (b) 3853 (c) 3848 (d) 3842
20. 20.What is the units digit of 9^12? (a) 9 (b) 2 (c) 1 (d) 0
21. 21.What is the smallest number you must add to 579 to make it divisible by 12? (a) 3 (b) 10 (c) 9 (d) 8
22. 22.What is the units digit of 9^24? (a) 2 (b) 1 (c) 9 (d) 0
23. 23.What is the units digit of 8^10? (a) 4 (b) 8 (c) 6 (d) 2
24. 24.Which of these numbers is divisible by 9? (a) 4621 (b) 4626 (c) 4633 (d) 4642
25. 25.Which of these numbers is divisible by 11? (a) 2401 (b) 2415 (c) 2394 (d) 2398
26. 26.What is the smallest number you must add to 253 to make it divisible by 10? (a) 3 (b) 8 (c) 6 (d) 7
27. 27.Which of these numbers is divisible by 3? (a) 1990 (b) 1985 (c) 1988 (d) 1986
28. 28.Which of these numbers is divisible by 3? (a) 301 (b) 299 (c) 300 (d) 305
29. 29.Which of these numbers is divisible by 9? (a) 5223 (b) 5220 (c) 5232 (d) 5219
30. 30.What is the remainder when 94 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
31. 31.What is the units digit of 7^57? (a) 7 (b) 1 (c) 3 (d) 9
32. 32.Which of these numbers is divisible by 11? (a) 4124 (b) 4096 (c) 4103 (d) 4107
33. 33.What is the units digit of 8^20? (a) 6 (b) 4 (c) 2 (d) 8
34. 34.What is the smallest number you must add to 342 to make it divisible by 7? (a) 3 (b) 6 (c) 1 (d) 2
35. 35.Which of these numbers is divisible by 9? (a) 2124 (b) 2129 (c) 2118 (d) 2135
36. 36.What is the units digit of 2^21? (a) 2 (b) 4 (c) 8 (d) 6
37. 37.What is the remainder when 300 is divided by 8? (a) 5 (b) 4 (c) 3 (d) 6
38. 38.Which of these numbers is divisible by 3? (a) 1094 (b) 1099 (c) 1096 (d) 1095
39. 39.What is the units digit of 7^22? (a) 3 (b) 7 (c) 1 (d) 9
40. 40.What is the smallest number you must add to 407 to make it divisible by 8? (a) 2 (b) 7 (c) 3 (d) 1

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