Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 844 · 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 8^31? (a) 2 (b) 8 (c) 6 (d) 4
2. 2.Which of these numbers is divisible by 4? (a) 2516 (b) 2514 (c) 2523 (d) 2519
3. 3.What is the smallest number you must add to 245 to make it divisible by 4? (a) 1 (b) 2 (c) 3
4. 4.Which of these numbers is divisible by 3? (a) 1166 (b) 1169 (c) 1167 (d) 1171
5. 5.What is the smallest number you must add to 102 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
6. 6.What is the smallest number you must add to 50 to make it divisible by 4? (a) 2 (b) 1 (c) 3
7. 7.Which of these numbers is divisible by 4? (a) 2483 (b) 2475 (c) 2479 (d) 2476
8. 8.What is the units digit of 4^33? (a) 4 (b) 6 (c) 5 (d) 3
9. 9.What is the remainder when 97 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
10. 10.Which of these numbers is divisible by 9? (a) 4655 (b) 4669 (c) 4651 (d) 4653
11. 11.What is the smallest number you must add to 128 to make it divisible by 10? (a) 2 (b) 3 (c) 8 (d) 1
12. 12.Which of these numbers is divisible by 4? (a) 591 (b) 584 (c) 583 (d) 587
13. 13.What is the remainder when 194 is divided by 6? (a) 4 (b) 2 (c) 3 (d) 1
14. 14.What is the smallest number you must add to 159 to make it divisible by 4? (a) 1 (b) 3 (c) 2
15. 15.Which of these numbers is divisible by 3? (a) 736 (b) 733 (c) 730 (d) 732
16. 16.Which of these numbers is divisible by 4? (a) 555 (b) 548 (c) 545 (d) 551
17. 17.What is the smallest number you must add to 672 to make it divisible by 9? (a) 3 (b) 6 (c) 2 (d) 4
18. 18.Which of these numbers is divisible by 3? (a) 1652 (b) 1649 (c) 1655 (d) 1650
19. 19.What is the remainder when 228 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
20. 20.What is the remainder when 381 is divided by 5? (a) 0 (b) 1 (c) 2 (d) 4
21. 21.What is the remainder when 184 is divided by 3? (a) 2 (b) 1 (c) 0
22. 22.What is the smallest number you must add to 136 to make it divisible by 9? (a) 1 (b) 7 (c) 6 (d) 8
23. 23.What is the smallest number you must add to 260 to make it divisible by 9? (a) 8 (b) 2 (c) 3 (d) 1
24. 24.What is the remainder when 190 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
25. 25.What is the remainder when 729 is divided by 11? (a) 3 (b) 8 (c) 2 (d) 4
26. 26.What is the units digit of 2^21? (a) 4 (b) 2 (c) 8 (d) 6
27. 27.What is the smallest number you must add to 475 to make it divisible by 10? (a) 4 (b) 7 (c) 6 (d) 5
28. 28.Which of these numbers is divisible by 4? (a) 2055 (b) 2050 (c) 2057 (d) 2052
29. 29.Which of these numbers is divisible by 8? (a) 6045 (b) 6032 (c) 6037 (d) 6031
30. 30.What is the remainder when 203 is divided by 3? (a) 1 (b) 2 (c) 0
31. 31.Which of these numbers is divisible by 4? (a) 314 (b) 311 (c) 319 (d) 312
32. 32.What is the remainder when 121 is divided by 9? (a) 3 (b) 4 (c) 6 (d) 5
33. 33.What is the remainder when 119 is divided by 8? (a) 6 (b) 1 (c) 7 (d) 5
34. 34.What is the remainder when 275 is divided by 10? (a) 4 (b) 7 (c) 5 (d) 6
35. 35.What is the units digit of 7^20? (a) 9 (b) 1 (c) 7 (d) 3
36. 36.What is the remainder when 230 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
37. 37.What is the remainder when 140 is divided by 12? (a) 4 (b) 7 (c) 8 (d) 9
38. 38.What is the remainder when 313 is divided by 6? (a) 5 (b) 1 (c) 0 (d) 2
39. 39.What is the smallest number you must add to 351 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
40. 40.Which of these numbers is divisible by 6? (a) 5223 (b) 5227 (c) 5217 (d) 5220

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