Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 832 · 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 2^46? (a) 6 (b) 4 (c) 8 (d) 2
2. 2.What is the remainder when 347 is divided by 7? (a) 6 (b) 4 (c) 3 (d) 5
3. 3.What is the units digit of 9^40? (a) 0 (b) 9 (c) 1 (d) 2
4. 4.What is the units digit of 9^14? (a) 0 (b) 9 (c) 2 (d) 1
5. 5.Which of these numbers is divisible by 3? (a) 457 (b) 461 (c) 459 (d) 464
6. 6.What is the remainder when 315 is divided by 10? (a) 6 (b) 7 (c) 4 (d) 5
7. 7.Which of these numbers is divisible by 9? (a) 2535 (b) 2543 (c) 2529 (d) 2524
8. 8.What is the units digit of 9^52? (a) 2 (b) 9 (c) 1 (d) 0
9. 9.What is the remainder when 485 is divided by 11? (a) 2 (b) 0 (c) 1 (d) 10
10. 10.What is the smallest number you must add to 180 to make it divisible by 11? (a) 4 (b) 6 (c) 8 (d) 7
11. 11.What is the remainder when 137 is divided by 5? (a) 4 (b) 3 (c) 1 (d) 2
12. 12.What is the smallest number you must add to 566 to make it divisible by 9? (a) 8 (b) 3 (c) 1 (d) 2
13. 13.What is the units digit of 3^35? (a) 9 (b) 1 (c) 7 (d) 3
14. 14.Which of these numbers is divisible by 3? (a) 268 (b) 266 (c) 262 (d) 264
15. 15.What is the remainder when 392 is divided by 6? (a) 3 (b) 1 (c) 4 (d) 2
16. 16.Which of these numbers is divisible by 8? (a) 6611 (b) 6607 (c) 6608 (d) 6623
17. 17.What is the remainder when 111 is divided by 5? (a) 2 (b) 4 (c) 0 (d) 1
18. 18.What is the remainder when 414 is divided by 8? (a) 7 (b) 2 (c) 6 (d) 5
19. 19.What is the units digit of 3^37? (a) 9 (b) 3 (c) 1 (d) 7
20. 20.Which of these numbers is divisible by 4? (a) 967 (b) 970 (c) 968 (d) 974
21. 21.What is the remainder when 143 is divided by 3? (a) 2 (b) 1 (c) 0
22. 22.Which of these numbers is divisible by 3? (a) 2180 (b) 2183 (c) 2181 (d) 2186
23. 23.Which of these numbers is divisible by 3? (a) 2108 (b) 2104 (c) 2106 (d) 2111
24. 24.What is the remainder when 126 is divided by 12? (a) 7 (b) 5 (c) 6 (d) 8
25. 25.Which of these numbers is divisible by 6? (a) 3781 (b) 3789 (c) 3776 (d) 3780
26. 26.Which of these numbers is divisible by 3? (a) 2463 (b) 2465 (c) 2461 (d) 2467
27. 27.Which of these numbers is divisible by 6? (a) 3344 (b) 3349 (c) 3342 (d) 3341
28. 28.What is the units digit of 9^10? (a) 9 (b) 2 (c) 0 (d) 1
29. 29.What is the smallest number you must add to 381 to make it divisible by 7? (a) 3 (b) 6 (c) 5 (d) 4
30. 30.What is the remainder when 91 is divided by 3? (a) 2 (b) 1 (c) 0
31. 31.What is the units digit of 3^48? (a) 9 (b) 3 (c) 1 (d) 7
32. 32.Which of these numbers is divisible by 8? (a) 1811 (b) 1808 (c) 1804 (d) 1823
33. 33.What is the units digit of 2^19? (a) 6 (b) 4 (c) 2 (d) 8
34. 34.What is the remainder when 98 is divided by 6? (a) 4 (b) 3 (c) 2 (d) 1
35. 35.What is the units digit of 2^18? (a) 6 (b) 4 (c) 2 (d) 8
36. 36.What is the smallest number you must add to 350 to make it divisible by 6? (a) 3 (b) 4 (c) 2 (d) 5
37. 37.What is the remainder when 106 is divided by 5? (a) 0 (b) 2 (c) 4 (d) 1
38. 38.Which of these numbers is divisible by 3? (a) 2537 (b) 2539 (c) 2535 (d) 2533
39. 39.What is the remainder when 479 is divided by 9? (a) 1 (b) 7 (c) 3 (d) 2
40. 40.What is the remainder when 88 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3

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