Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 549 · 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 4^58? (a) 7 (b) 4 (c) 5 (d) 6
2. 2.What is the remainder when 134 is divided by 3? (a) 0 (b) 1 (c) 2
3. 3.What is the remainder when 575 is divided by 10? (a) 7 (b) 4 (c) 5 (d) 6
4. 4.Which of these numbers is divisible by 11? (a) 4420 (b) 4430 (c) 4409 (d) 4411
5. 5.What is the remainder when 592 is divided by 10? (a) 3 (b) 1 (c) 8 (d) 2
6. 6.What is the remainder when 209 is divided by 8? (a) 0 (b) 7 (c) 2 (d) 1
7. 7.What is the units digit of 7^40? (a) 9 (b) 7 (c) 1 (d) 3
8. 8.What is the smallest number you must add to 194 to make it divisible by 4? (a) 1 (b) 3 (c) 2
9. 9.Which of these numbers is divisible by 6? (a) 3597 (b) 3592 (c) 3594 (d) 3605
10. 10.What is the smallest number you must add to 347 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
11. 11.What is the remainder when 388 is divided by 7? (a) 4 (b) 2 (c) 5 (d) 3
12. 12.What is the units digit of 4^11? (a) 3 (b) 4 (c) 5 (d) 6
13. 13.What is the smallest number you must add to 596 to make it divisible by 9? (a) 8 (b) 2 (c) 6 (d) 7
14. 14.What is the units digit of 8^15? (a) 8 (b) 2 (c) 4 (d) 6
15. 15.What is the remainder when 494 is divided by 12? (a) 10 (b) 3 (c) 2 (d) 1
16. 16.What is the remainder when 805 is divided by 12? (a) 11 (b) 1 (c) 2 (d) 0
17. 17.What is the remainder when 166 is divided by 6? (a) 4 (b) 5 (c) 2 (d) 3
18. 18.What is the units digit of 7^14? (a) 7 (b) 9 (c) 3 (d) 1
19. 19.What is the units digit of 8^51? (a) 8 (b) 6 (c) 4 (d) 2
20. 20.What is the remainder when 207 is divided by 10? (a) 7 (b) 8 (c) 3 (d) 6
21. 21.What is the remainder when 238 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
22. 22.What is the units digit of 2^37? (a) 2 (b) 8 (c) 4 (d) 6
23. 23.What is the smallest number you must add to 606 to make it divisible by 12? (a) 5 (b) 7 (c) 6 (d) 8
24. 24.What is the smallest number you must add to 227 to make it divisible by 7? (a) 4 (b) 6 (c) 5 (d) 3
25. 25.What is the units digit of 9^25? (a) 9 (b) 7 (c) 8 (d) 1
26. 26.Which of these numbers is divisible by 9? (a) 3654 (b) 3655 (c) 3665 (d) 3649
27. 27.Which of these numbers is divisible by 8? (a) 4415 (b) 4408 (c) 4419 (d) 4402
28. 28.Which of these numbers is divisible by 4? (a) 170 (b) 177 (c) 175 (d) 172
29. 29.What is the remainder when 165 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
30. 30.What is the remainder when 426 is divided by 11? (a) 7 (b) 3 (c) 9 (d) 8
31. 31.What is the remainder when 159 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
32. 32.Which of these numbers is divisible by 6? (a) 5136 (b) 5135 (c) 5143 (d) 5140
33. 33.What is the smallest number you must add to 67 to make it divisible by 4? (a) 3 (b) 2 (c) 1
34. 34.What is the smallest number you must add to 278 to make it divisible by 10? (a) 2 (b) 8 (c) 1 (d) 3
35. 35.What is the smallest number you must add to 266 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
36. 36.What is the remainder when 293 is divided by 9? (a) 4 (b) 6 (c) 5 (d) 7
37. 37.What is the remainder when 65 is divided by 3? (a) 1 (b) 0 (c) 2
38. 38.What is the units digit of 8^53? (a) 4 (b) 6 (c) 8 (d) 2
39. 39.What is the smallest number you must add to 478 to make it divisible by 7? (a) 6 (b) 2 (c) 5 (d) 4
40. 40.What is the units digit of 7^32? (a) 3 (b) 1 (c) 7 (d) 9

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