Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 761 · 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 7^58? (a) 9 (b) 1 (c) 3 (d) 7
2. 2.What is the remainder when 431 is divided by 7? (a) 4 (b) 3 (c) 6 (d) 5
3. 3.Which of these numbers is divisible by 11? (a) 1419 (b) 1422 (c) 1439 (d) 1411
4. 4.What is the smallest number you must add to 172 to make it divisible by 3? (a) 2 (b) 1
5. 5.What is the remainder when 838 is divided by 11? (a) 9 (b) 1 (c) 2 (d) 3
6. 6.Which of these numbers is divisible by 11? (a) 2826 (b) 2831 (c) 2816 (d) 2812
7. 7.Which of these numbers is divisible by 3? (a) 932 (b) 928 (c) 934 (d) 930
8. 8.What is the smallest number you must add to 580 to make it divisible by 12? (a) 8 (b) 4 (c) 9 (d) 7
9. 9.What is the remainder when 399 is divided by 10? (a) 7 (b) 8 (c) 9 (d) 1
10. 10.What is the remainder when 526 is divided by 7? (a) 1 (b) 2 (c) 6 (d) 0
11. 11.What is the smallest number you must add to 105 to make it divisible by 6? (a) 2 (b) 4 (c) 3 (d) 5
12. 12.What is the units digit of 2^17? (a) 8 (b) 6 (c) 2 (d) 4
13. 13.Which of these numbers is divisible by 6? (a) 930 (b) 925 (c) 934 (d) 937
14. 14.What is the units digit of 8^47? (a) 8 (b) 4 (c) 6 (d) 2
15. 15.Which of these numbers is divisible by 11? (a) 8034 (b) 8023 (c) 8016 (d) 8019
16. 16.What is the remainder when 415 is divided by 7? (a) 2 (b) 3 (c) 1 (d) 5
17. 17.What is the smallest number you must add to 116 to make it divisible by 8? (a) 6 (b) 3 (c) 4 (d) 5
18. 18.What is the smallest number you must add to 160 to make it divisible by 3? (a) 2 (b) 1
19. 19.Which of these numbers is divisible by 4? (a) 1351 (b) 1344 (c) 1343 (d) 1345
20. 20.What is the units digit of 9^19? (a) 9 (b) 1 (c) 8 (d) 7
21. 21.What is the units digit of 9^43? (a) 1 (b) 8 (c) 9 (d) 7
22. 22.What is the smallest number you must add to 814 to make it divisible by 12? (a) 1 (b) 3 (c) 2 (d) 10
23. 23.What is the units digit of 2^27? (a) 8 (b) 2 (c) 4 (d) 6
24. 24.What is the remainder when 739 is divided by 11? (a) 2 (b) 3 (c) 9 (d) 1
25. 25.What is the units digit of 2^53? (a) 8 (b) 6 (c) 2 (d) 4
26. 26.What is the units digit of 4^14? (a) 4 (b) 5 (c) 7 (d) 6
27. 27.What is the remainder when 497 is divided by 9? (a) 3 (b) 7 (c) 1 (d) 2
28. 28.What is the remainder when 53 is divided by 3? (a) 0 (b) 1 (c) 2
29. 29.Which of these numbers is divisible by 11? (a) 8516 (b) 8530 (c) 8525 (d) 8546
30. 30.Which of these numbers is divisible by 4? (a) 2777 (b) 2775 (c) 2770 (d) 2772
31. 31.What is the remainder when 430 is divided by 8? (a) 2 (b) 5 (c) 6 (d) 7
32. 32.What is the remainder when 535 is divided by 7? (a) 4 (b) 5 (c) 3 (d) 2
33. 33.What is the remainder when 548 is divided by 7? (a) 1 (b) 2 (c) 5 (d) 3
34. 34.Which of these numbers is divisible by 4? (a) 2448 (b) 2450 (c) 2446 (d) 2454
35. 35.Which of these numbers is divisible by 6? (a) 1826 (b) 1816 (c) 1819 (d) 1818
36. 36.What is the smallest number you must add to 389 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
37. 37.What is the units digit of 4^32? (a) 7 (b) 6 (c) 5 (d) 4
38. 38.What is the units digit of 7^52? (a) 9 (b) 3 (c) 7 (d) 1
39. 39.What is the remainder when 111 is divided by 9? (a) 2 (b) 3 (c) 4 (d) 6
40. 40.Which of these numbers is divisible by 11? (a) 6897 (b) 6894 (c) 6912 (d) 6902

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