Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 189 · 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.Which of these numbers is divisible by 6? (a) 1179 (b) 1176 (c) 1186 (d) 1174
2. 2.What is the remainder when 176 is divided by 10? (a) 4 (b) 6 (c) 7 (d) 5
3. 3.Which of these numbers is divisible by 6? (a) 1224 (b) 1223 (c) 1231 (d) 1227
4. 4.What is the smallest number you must add to 238 to make it divisible by 3? (a) 1 (b) 2
5. 5.Which of these numbers is divisible by 3? (a) 1505 (b) 1501 (c) 1500 (d) 1499
6. 6.Which of these numbers is divisible by 6? (a) 1293 (b) 1306 (c) 1296 (d) 1299
7. 7.What is the smallest number you must add to 429 to make it divisible by 12? (a) 9 (b) 2 (c) 3 (d) 4
8. 8.Which of these numbers is divisible by 4? (a) 365 (b) 369 (c) 368 (d) 375
9. 9.Which of these numbers is divisible by 4? (a) 2722 (b) 2716 (c) 2718 (d) 2715
10. 10.What is the remainder when 161 is divided by 3? (a) 0 (b) 1 (c) 2
11. 11.Which of these numbers is divisible by 11? (a) 4002 (b) 3993 (c) 3983 (d) 4008
12. 12.Which of these numbers is divisible by 6? (a) 3642 (b) 3646 (c) 3651 (d) 3639
13. 13.What is the remainder when 443 is divided by 6? (a) 4 (b) 3 (c) 1 (d) 5
14. 14.What is the remainder when 151 is divided by 6? (a) 0 (b) 1 (c) 5 (d) 2
15. 15.Which of these numbers is divisible by 6? (a) 2263 (b) 2255 (c) 2260 (d) 2256
16. 16.Which of these numbers is divisible by 8? (a) 4166 (b) 4175 (c) 4160 (d) 4158
17. 17.Which of these numbers is divisible by 6? (a) 1730 (b) 1743 (c) 1734 (d) 1737
18. 18.What is the remainder when 569 is divided by 10? (a) 1 (b) 8 (c) 7 (d) 9
19. 19.Which of these numbers is divisible by 9? (a) 4161 (b) 4167 (c) 4181 (d) 4173
20. 20.Which of these numbers is divisible by 9? (a) 4716 (b) 4723 (c) 4710 (d) 4730
21. 21.What is the remainder when 611 is divided by 11? (a) 5 (b) 7 (c) 6 (d) 8
22. 22.What is the smallest number you must add to 148 to make it divisible by 7? (a) 5 (b) 1 (c) 4 (d) 6
23. 23.What is the remainder when 264 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
24. 24.Which of these numbers is divisible by 9? (a) 481 (b) 464 (c) 473 (d) 468
25. 25.What is the remainder when 328 is divided by 10? (a) 2 (b) 9 (c) 8 (d) 7
26. 26.Which of these numbers is divisible by 9? (a) 2058 (b) 2062 (c) 2044 (d) 2052
27. 27.Which of these numbers is divisible by 11? (a) 3674 (b) 3690 (c) 3682 (d) 3672
28. 28.What is the remainder when 294 is divided by 9? (a) 6 (b) 3 (c) 7 (d) 5
29. 29.What is the remainder when 80 is divided by 3? (a) 1 (b) 0 (c) 2
30. 30.What is the units digit of 9^54? (a) 9 (b) 2 (c) 0 (d) 1
31. 31.What is the smallest number you must add to 111 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
32. 32.What is the remainder when 297 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
33. 33.What is the units digit of 7^33? (a) 1 (b) 7 (c) 3 (d) 9
34. 34.What is the smallest number you must add to 673 to make it divisible by 11? (a) 10 (b) 8 (c) 2 (d) 9
35. 35.What is the units digit of 3^10? (a) 1 (b) 3 (c) 7 (d) 9
36. 36.What is the units digit of 3^40? (a) 9 (b) 3 (c) 7 (d) 1
37. 37.What is the remainder when 343 is divided by 10? (a) 7 (b) 4 (c) 3 (d) 2
38. 38.Which of these numbers is divisible by 11? (a) 5542 (b) 5533 (c) 5531 (d) 5550
39. 39.What is the smallest number you must add to 116 to make it divisible by 3? (a) 2 (b) 1
40. 40.What is the remainder when 202 is divided by 3? (a) 0 (b) 2 (c) 1

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