Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 480 · 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 97 is divided by 9? (a) 8 (b) 7 (c) 2 (d) 6
2. 2.What is the smallest number you must add to 210 to make it divisible by 4? (a) 1 (b) 2 (c) 3
3. 3.What is the smallest number you must add to 697 to make it divisible by 12? (a) 1 (b) 11 (c) 10 (d) 9
4. 4.What is the remainder when 162 is divided by 8? (a) 2 (b) 1 (c) 6 (d) 3
5. 5.Which of these numbers is divisible by 4? (a) 1024 (b) 1030 (c) 1025 (d) 1022
6. 6.What is the smallest number you must add to 447 to make it divisible by 6? (a) 4 (b) 3 (c) 5 (d) 2
7. 7.What is the units digit of 3^16? (a) 7 (b) 9 (c) 1 (d) 3
8. 8.What is the smallest number you must add to 158 to make it divisible by 4? (a) 3 (b) 2 (c) 1
9. 9.Which of these numbers is divisible by 3? (a) 2201 (b) 2197 (c) 2199 (d) 2203
10. 10.What is the remainder when 474 is divided by 11? (a) 0 (b) 1 (c) 2 (d) 10
11. 11.What is the smallest number you must add to 253 to make it divisible by 9? (a) 8 (b) 7 (c) 6 (d) 1
12. 12.What is the units digit of 4^57? (a) 3 (b) 4 (c) 5 (d) 6
13. 13.Which of these numbers is divisible by 11? (a) 6558 (b) 6584 (c) 6573 (d) 6567
14. 14.What is the remainder when 45 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
15. 15.What is the units digit of 4^33? (a) 4 (b) 3 (c) 5 (d) 6
16. 16.What is the smallest number you must add to 318 to make it divisible by 8? (a) 2 (b) 1 (c) 3 (d) 6
17. 17.What is the units digit of 7^43? (a) 1 (b) 7 (c) 9 (d) 3
18. 18.Which of these numbers is divisible by 3? (a) 1953 (b) 1952 (c) 1954 (d) 1957
19. 19.What is the units digit of 4^37? (a) 5 (b) 3 (c) 6 (d) 4
20. 20.What is the remainder when 505 is divided by 7? (a) 6 (b) 1 (c) 0 (d) 2
21. 21.What is the remainder when 124 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
22. 22.What is the units digit of 9^21? (a) 1 (b) 9 (c) 7 (d) 8
23. 23.What is the units digit of 3^51? (a) 1 (b) 3 (c) 7 (d) 9
24. 24.Which of these numbers is divisible by 9? (a) 918 (b) 916 (c) 922 (d) 935
25. 25.Which of these numbers is divisible by 11? (a) 464 (b) 491 (c) 481 (d) 473
26. 26.What is the remainder when 470 is divided by 11? (a) 9 (b) 7 (c) 8 (d) 3
27. 27.What is the units digit of 7^33? (a) 3 (b) 7 (c) 1 (d) 9
28. 28.Which of these numbers is divisible by 4? (a) 3238 (b) 3236 (c) 3243 (d) 3233
29. 29.What is the smallest number you must add to 588 to make it divisible by 8? (a) 5 (b) 6 (c) 4 (d) 3
30. 30.What is the units digit of 7^17? (a) 1 (b) 3 (c) 9 (d) 7
31. 31.What is the remainder when 166 is divided by 3? (a) 2 (b) 0 (c) 1
32. 32.What is the remainder when 209 is divided by 6? (a) 4 (b) 3 (c) 5 (d) 1
33. 33.What is the remainder when 79 is divided by 3? (a) 2 (b) 0 (c) 1
34. 34.Which of these numbers is divisible by 6? (a) 4163 (b) 4156 (c) 4152 (d) 4147
35. 35.Which of these numbers is divisible by 4? (a) 3042 (b) 3046 (c) 3044 (d) 3050
36. 36.What is the smallest number you must add to 297 to make it divisible by 4? (a) 1 (b) 3 (c) 2
37. 37.What is the units digit of 3^36? (a) 9 (b) 3 (c) 1 (d) 7
38. 38.What is the remainder when 183 is divided by 6? (a) 2 (b) 5 (c) 4 (d) 3
39. 39.What is the remainder when 331 is divided by 6? (a) 5 (b) 2 (c) 1 (d) 0
40. 40.Which of these numbers is divisible by 4? (a) 2664 (b) 2671 (c) 2666 (d) 2661

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