Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 87 · 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 remainder when 161 is divided by 9? (a) 6 (b) 8 (c) 7 (d) 1
2. 2.What is the remainder when 122 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
3. 3.What is the remainder when 369 is divided by 12? (a) 8 (b) 10 (c) 3 (d) 9
4. 4.Which of these numbers is divisible by 6? (a) 1925 (b) 1929 (c) 1936 (d) 1926
5. 5.What is the remainder when 359 is divided by 11? (a) 8 (b) 4 (c) 6 (d) 7
6. 6.What is the units digit of 4^15? (a) 6 (b) 3 (c) 4 (d) 5
7. 7.What is the remainder when 455 is divided by 9? (a) 4 (b) 7 (c) 5 (d) 6
8. 8.What is the units digit of 3^21? (a) 9 (b) 7 (c) 3 (d) 1
9. 9.What is the units digit of 2^52? (a) 8 (b) 6 (c) 2 (d) 4
10. 10.What is the units digit of 9^49? (a) 1 (b) 7 (c) 9 (d) 8
11. 11.Which of these numbers is divisible by 6? (a) 673 (b) 671 (c) 683 (d) 672
12. 12.What is the smallest number you must add to 221 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
13. 13.What is the remainder when 214 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
14. 14.What is the units digit of 2^18? (a) 8 (b) 6 (c) 4 (d) 2
15. 15.What is the smallest number you must add to 136 to make it divisible by 3? (a) 1 (b) 2
16. 16.What is the remainder when 45 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
17. 17.Which of these numbers is divisible by 9? (a) 5603 (b) 5590 (c) 5587 (d) 5589
18. 18.What is the remainder when 406 is divided by 8? (a) 2 (b) 7 (c) 5 (d) 6
19. 19.What is the units digit of 2^46? (a) 2 (b) 4 (c) 6 (d) 8
20. 20.Which of these numbers is divisible by 9? (a) 3798 (b) 3803 (c) 3809 (d) 3795
21. 21.What is the remainder when 61 is divided by 3? (a) 0 (b) 2 (c) 1
22. 22.Which of these numbers is divisible by 8? (a) 492 (b) 500 (c) 486 (d) 488
23. 23.What is the remainder when 402 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
24. 24.What is the units digit of 2^36? (a) 4 (b) 6 (c) 8 (d) 2
25. 25.What is the smallest number you must add to 186 to make it divisible by 12? (a) 6 (b) 7 (c) 5 (d) 8
26. 26.What is the units digit of 9^15? (a) 1 (b) 7 (c) 8 (d) 9
27. 27.What is the units digit of 2^22? (a) 8 (b) 6 (c) 2 (d) 4
28. 28.What is the units digit of 7^44? (a) 9 (b) 3 (c) 1 (d) 7
29. 29.What is the units digit of 9^23? (a) 1 (b) 7 (c) 8 (d) 9
30. 30.What is the remainder when 265 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
31. 31.What is the remainder when 293 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
32. 32.What is the smallest number you must add to 94 to make it divisible by 7? (a) 3 (b) 5 (c) 4 (d) 6
33. 33.What is the smallest number you must add to 297 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
34. 34.What is the units digit of 8^48? (a) 6 (b) 8 (c) 4 (d) 2
35. 35.What is the units digit of 9^59? (a) 9 (b) 1 (c) 8 (d) 7
36. 36.Which of these numbers is divisible by 3? (a) 1520 (b) 1523 (c) 1521 (d) 1525
37. 37.What is the remainder when 359 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
38. 38.What is the units digit of 4^36? (a) 4 (b) 6 (c) 7 (d) 5
39. 39.What is the smallest number you must add to 205 to make it divisible by 4? (a) 2 (b) 3 (c) 1
40. 40.What is the remainder when 493 is divided by 10? (a) 4 (b) 7 (c) 3 (d) 2

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