Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 907 · 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 746 is divided by 10? (a) 4 (b) 7 (c) 5 (d) 6
2. 2.What is the remainder when 340 is divided by 8? (a) 3 (b) 4 (c) 6 (d) 5
3. 3.Which of these numbers is divisible by 6? (a) 4964 (b) 4961 (c) 4972 (d) 4962
4. 4.Which of these numbers is divisible by 6? (a) 4589 (b) 4578 (c) 4582 (d) 4573
5. 5.What is the remainder when 208 is divided by 9? (a) 8 (b) 2 (c) 1 (d) 0
6. 6.Which of these numbers is divisible by 8? (a) 4069 (b) 4086 (c) 4077 (d) 4072
7. 7.What is the units digit of 8^31? (a) 8 (b) 4 (c) 2 (d) 6
8. 8.Which of these numbers is divisible by 8? (a) 1236 (b) 1232 (c) 1247 (d) 1225
9. 9.Which of these numbers is divisible by 11? (a) 2959 (b) 2974 (c) 2949 (d) 2966
10. 10.What is the smallest number you must add to 91 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
11. 11.What is the smallest number you must add to 187 to make it divisible by 3? (a) 1 (b) 2
12. 12.Which of these numbers is divisible by 9? (a) 3104 (b) 3118 (c) 3105 (d) 3109
13. 13.Which of these numbers is divisible by 4? (a) 2956 (b) 2962 (c) 2955 (d) 2959
14. 14.Which of these numbers is divisible by 6? (a) 4181 (b) 4191 (c) 4182 (d) 4186
15. 15.What is the units digit of 7^17? (a) 1 (b) 3 (c) 7 (d) 9
16. 16.What is the smallest number you must add to 44 to make it divisible by 3? (a) 2 (b) 1
17. 17.What is the smallest number you must add to 617 to make it divisible by 8? (a) 5 (b) 6 (c) 7 (d) 1
18. 18.What is the remainder when 233 is divided by 3? (a) 0 (b) 1 (c) 2
19. 19.What is the remainder when 266 is divided by 9? (a) 7 (b) 4 (c) 5 (d) 6
20. 20.What is the remainder when 704 is divided by 12? (a) 4 (b) 7 (c) 9 (d) 8
21. 21.What is the units digit of 4^49? (a) 5 (b) 6 (c) 3 (d) 4
22. 22.Which of these numbers is divisible by 9? (a) 2938 (b) 2949 (c) 2959 (d) 2943
23. 23.What is the remainder when 287 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
24. 24.What is the units digit of 8^36? (a) 8 (b) 4 (c) 6 (d) 2
25. 25.Which of these numbers is divisible by 4? (a) 3553 (b) 3558 (c) 3556 (d) 3563
26. 26.What is the smallest number you must add to 54 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
27. 27.What is the smallest number you must add to 677 to make it divisible by 9? (a) 7 (b) 8 (c) 6 (d) 2
28. 28.What is the units digit of 8^19? (a) 4 (b) 6 (c) 8 (d) 2
29. 29.What is the smallest number you must add to 320 to make it divisible by 11? (a) 9 (b) 8 (c) 10 (d) 1
30. 30.Which of these numbers is divisible by 6? (a) 1682 (b) 1674 (c) 1672 (d) 1679
31. 31.Which of these numbers is divisible by 4? (a) 1423 (b) 1414 (c) 1416 (d) 1418
32. 32.What is the remainder when 74 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
33. 33.Which of these numbers is divisible by 9? (a) 5265 (b) 5276 (c) 5258 (d) 5267
34. 34.Which of these numbers is divisible by 11? (a) 3191 (b) 3179 (c) 3182 (d) 3174
35. 35.What is the smallest number you must add to 436 to make it divisible by 6? (a) 4 (b) 1 (c) 2 (d) 3
36. 36.Which of these numbers is divisible by 11? (a) 6369 (b) 6368 (c) 6372 (d) 6381
37. 37.What is the units digit of 8^58? (a) 2 (b) 4 (c) 8 (d) 6
38. 38.What is the units digit of 2^48? (a) 8 (b) 4 (c) 6 (d) 2
39. 39.Which of these numbers is divisible by 11? (a) 8874 (b) 8879 (c) 8877 (d) 8893
40. 40.What is the remainder when 73 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3

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