Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 35 · 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 7^25? (a) 1 (b) 9 (c) 7 (d) 3
2. 2.What is the units digit of 4^38? (a) 5 (b) 7 (c) 4 (d) 6
3. 3.What is the remainder when 202 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
4. 4.Which of these numbers is divisible by 8? (a) 3482 (b) 3480 (c) 3492 (d) 3473
5. 5.Which of these numbers is divisible by 3? (a) 704 (b) 699 (c) 697 (d) 700
6. 6.Which of these numbers is divisible by 6? (a) 2871 (b) 2862 (c) 2865 (d) 2861
7. 7.What is the units digit of 9^42? (a) 1 (b) 0 (c) 9 (d) 2
8. 8.What is the remainder when 104 is divided by 3? (a) 1 (b) 0 (c) 2
9. 9.What is the smallest number you must add to 138 to make it divisible by 10? (a) 3 (b) 8 (c) 2 (d) 1
10. 10.What is the remainder when 82 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
11. 11.What is the smallest number you must add to 166 to make it divisible by 11? (a) 1 (b) 9 (c) 10 (d) 8
12. 12.What is the units digit of 9^33? (a) 1 (b) 7 (c) 9 (d) 8
13. 13.What is the remainder when 412 is divided by 7? (a) 1 (b) 5 (c) 4 (d) 6
14. 14.What is the remainder when 604 is divided by 8? (a) 6 (b) 5 (c) 3 (d) 4
15. 15.What is the units digit of 2^24? (a) 4 (b) 2 (c) 8 (d) 6
16. 16.What is the remainder when 313 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
17. 17.What is the units digit of 3^18? (a) 9 (b) 7 (c) 3 (d) 1
18. 18.What is the smallest number you must add to 260 to make it divisible by 7? (a) 4 (b) 6 (c) 1 (d) 5
19. 19.What is the units digit of 7^46? (a) 1 (b) 7 (c) 3 (d) 9
20. 20.What is the remainder when 184 is divided by 7? (a) 5 (b) 2 (c) 3 (d) 1
21. 21.What is the remainder when 136 is divided by 3? (a) 1 (b) 0 (c) 2
22. 22.What is the smallest number you must add to 40 to make it divisible by 3? (a) 2 (b) 1
23. 23.Which of these numbers is divisible by 8? (a) 414 (b) 417 (c) 416 (d) 429
24. 24.Which of these numbers is divisible by 3? (a) 215 (b) 217 (c) 220 (d) 216
25. 25.Which of these numbers is divisible by 11? (a) 1507 (b) 1505 (c) 1516 (d) 1524
26. 26.Which of these numbers is divisible by 4? (a) 3570 (b) 3573 (c) 3566 (d) 3568
27. 27.Which of these numbers is divisible by 11? (a) 4266 (b) 4278 (c) 4285 (d) 4268
28. 28.Which of these numbers is divisible by 11? (a) 2941 (b) 2956 (c) 2948 (d) 2962
29. 29.What is the remainder when 133 is divided by 3? (a) 2 (b) 1 (c) 0
30. 30.Which of these numbers is divisible by 9? (a) 6723 (b) 6721 (c) 6735 (d) 6729
31. 31.Which of these numbers is divisible by 4? (a) 3351 (b) 3344 (c) 3341 (d) 3346
32. 32.Which of these numbers is divisible by 6? (a) 1548 (b) 1547 (c) 1558 (d) 1553
33. 33.What is the units digit of 7^55? (a) 7 (b) 1 (c) 9 (d) 3
34. 34.Which of these numbers is divisible by 8? (a) 4976 (b) 4983 (c) 4987 (d) 4970
35. 35.What is the units digit of 3^15? (a) 1 (b) 3 (c) 7 (d) 9
36. 36.What is the smallest number you must add to 145 to make it divisible by 12? (a) 10 (b) 9 (c) 1 (d) 11
37. 37.What is the units digit of 7^14? (a) 3 (b) 1 (c) 9 (d) 7
38. 38.What is the units digit of 2^58? (a) 4 (b) 2 (c) 6 (d) 8
39. 39.What is the units digit of 4^54? (a) 5 (b) 4 (c) 7 (d) 6
40. 40.What is the smallest number you must add to 886 to make it divisible by 11? (a) 5 (b) 6 (c) 7 (d) 4

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