Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 531 · 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 3? (a) 1485 (b) 1490 (c) 1484 (d) 1486
2. 2.What is the units digit of 3^31? (a) 3 (b) 1 (c) 9 (d) 7
3. 3.What is the remainder when 404 is divided by 7? (a) 6 (b) 2 (c) 4 (d) 5
4. 4.What is the units digit of 4^35? (a) 5 (b) 3 (c) 6 (d) 4
5. 5.What is the units digit of 9^17? (a) 1 (b) 9 (c) 8 (d) 7
6. 6.What is the remainder when 137 is divided by 8? (a) 7 (b) 2 (c) 1 (d) 0
7. 7.What is the remainder when 202 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
8. 8.What is the smallest number you must add to 118 to make it divisible by 8? (a) 3 (b) 1 (c) 2 (d) 6
9. 9.What is the smallest number you must add to 778 to make it divisible by 10? (a) 2 (b) 3 (c) 8 (d) 1
10. 10.What is the remainder when 112 is divided by 3? (a) 2 (b) 0 (c) 1
11. 11.What is the units digit of 9^51? (a) 1 (b) 9 (c) 8 (d) 7
12. 12.What is the smallest number you must add to 189 to make it divisible by 11? (a) 10 (b) 2 (c) 8 (d) 9
13. 13.What is the units digit of 8^18? (a) 4 (b) 2 (c) 8 (d) 6
14. 14.What is the remainder when 356 is divided by 10? (a) 5 (b) 7 (c) 6 (d) 4
15. 15.Which of these numbers is divisible by 6? (a) 343 (b) 334 (c) 336 (d) 340
16. 16.What is the smallest number you must add to 125 to make it divisible by 10? (a) 7 (b) 4 (c) 6 (d) 5
17. 17.What is the units digit of 2^16? (a) 6 (b) 4 (c) 2 (d) 8
18. 18.What is the smallest number you must add to 214 to make it divisible by 9? (a) 2 (b) 1 (c) 3 (d) 7
19. 19.What is the remainder when 337 is divided by 8? (a) 7 (b) 1 (c) 2 (d) 0
20. 20.What is the remainder when 91 is divided by 6? (a) 1 (b) 2 (c) 0 (d) 5
21. 21.What is the remainder when 202 is divided by 10? (a) 3 (b) 2 (c) 1 (d) 8
22. 22.What is the units digit of 9^19? (a) 1 (b) 9 (c) 8 (d) 7
23. 23.What is the remainder when 179 is divided by 3? (a) 0 (b) 1 (c) 2
24. 24.Which of these numbers is divisible by 8? (a) 2419 (b) 2410 (c) 2408 (d) 2401
25. 25.Which of these numbers is divisible by 4? (a) 2646 (b) 2643 (c) 2650 (d) 2644
26. 26.What is the smallest number you must add to 251 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
27. 27.Which of these numbers is divisible by 11? (a) 615 (b) 624 (c) 605 (d) 603
28. 28.What is the remainder when 259 is divided by 9? (a) 2 (b) 7 (c) 6 (d) 8
29. 29.Which of these numbers is divisible by 6? (a) 4158 (b) 4167 (c) 4157 (d) 4162
30. 30.What is the smallest number you must add to 185 to make it divisible by 6? (a) 5 (b) 3 (c) 1 (d) 2
31. 31.What is the remainder when 266 is divided by 8? (a) 3 (b) 2 (c) 1 (d) 6
32. 32.Which of these numbers is divisible by 9? (a) 402 (b) 390 (c) 380 (d) 387
33. 33.What is the units digit of 2^58? (a) 2 (b) 4 (c) 8 (d) 6
34. 34.What is the remainder when 158 is divided by 8? (a) 2 (b) 7 (c) 5 (d) 6
35. 35.What is the units digit of 7^53? (a) 3 (b) 9 (c) 7 (d) 1
36. 36.What is the remainder when 349 is divided by 9? (a) 6 (b) 7 (c) 8 (d) 2
37. 37.What is the remainder when 223 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
38. 38.What is the smallest number you must add to 217 to make it divisible by 3? (a) 1 (b) 2
39. 39.What is the remainder when 219 is divided by 9? (a) 2 (b) 3 (c) 4 (d) 6
40. 40.What is the smallest number you must add to 161 to make it divisible by 6? (a) 5 (b) 2 (c) 1 (d) 3

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