Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 572 · 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 7^12? (a) 9 (b) 7 (c) 3 (d) 1
2. 2.Which of these numbers is divisible by 4? (a) 2120 (b) 2117 (c) 2122 (d) 2126
3. 3.What is the remainder when 155 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
4. 4.What is the units digit of 9^41? (a) 1 (b) 8 (c) 7 (d) 9
5. 5.What is the remainder when 179 is divided by 3? (a) 2 (b) 1 (c) 0
6. 6.Which of these numbers is divisible by 11? (a) 1191 (b) 1200 (c) 1188 (d) 1184
7. 7.Which of these numbers is divisible by 11? (a) 7322 (b) 7313 (c) 7333 (d) 7315
8. 8.Which of these numbers is divisible by 6? (a) 3943 (b) 3951 (c) 3957 (d) 3948
9. 9.What is the smallest number you must add to 333 to make it divisible by 11? (a) 3 (b) 9 (c) 7 (d) 8
10. 10.What is the smallest number you must add to 199 to make it divisible by 4? (a) 3 (b) 2 (c) 1
11. 11.What is the remainder when 407 is divided by 8? (a) 1 (b) 5 (c) 6 (d) 7
12. 12.What is the units digit of 7^59? (a) 1 (b) 7 (c) 3 (d) 9
13. 13.What is the smallest number you must add to 85 to make it divisible by 6? (a) 1 (b) 4 (c) 5 (d) 3
14. 14.What is the remainder when 119 is divided by 3? (a) 0 (b) 2 (c) 1
15. 15.What is the smallest number you must add to 595 to make it divisible by 11? (a) 1 (b) 10 (c) 8 (d) 9
16. 16.What is the remainder when 95 is divided by 6? (a) 4 (b) 3 (c) 1 (d) 5
17. 17.What is the smallest number you must add to 110 to make it divisible by 4? (a) 1 (b) 3 (c) 2
18. 18.Which of these numbers is divisible by 8? (a) 3357 (b) 3340 (c) 3344 (d) 3346
19. 19.What is the units digit of 9^20? (a) 9 (b) 0 (c) 1 (d) 2
20. 20.What is the units digit of 3^49? (a) 7 (b) 9 (c) 1 (d) 3
21. 21.What is the remainder when 100 is divided by 3? (a) 0 (b) 2 (c) 1
22. 22.What is the smallest number you must add to 349 to make it divisible by 12? (a) 10 (b) 9 (c) 1 (d) 11
23. 23.What is the smallest number you must add to 179 to make it divisible by 8? (a) 6 (b) 5 (c) 4 (d) 3
24. 24.Which of these numbers is divisible by 8? (a) 3173 (b) 3160 (c) 3158 (d) 3163
25. 25.What is the remainder when 328 is divided by 9? (a) 5 (b) 4 (c) 6 (d) 3
26. 26.What is the smallest number you must add to 479 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 1
27. 27.What is the remainder when 761 is divided by 10? (a) 9 (b) 0 (c) 2 (d) 1
28. 28.Which of these numbers is divisible by 11? (a) 2445 (b) 2457 (c) 2442 (d) 2432
29. 29.What is the units digit of 3^33? (a) 7 (b) 1 (c) 3 (d) 9
30. 30.Which of these numbers is divisible by 8? (a) 1884 (b) 1888 (c) 1891 (d) 1902
31. 31.Which of these numbers is divisible by 4? (a) 792 (b) 797 (c) 789 (d) 794
32. 32.What is the units digit of 3^58? (a) 9 (b) 1 (c) 3 (d) 7
33. 33.What is the remainder when 49 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
34. 34.Which of these numbers is divisible by 4? (a) 2079 (b) 2080 (c) 2087 (d) 2083
35. 35.What is the remainder when 113 is divided by 6? (a) 3 (b) 1 (c) 5 (d) 4
36. 36.What is the units digit of 2^47? (a) 4 (b) 6 (c) 8 (d) 2
37. 37.What is the smallest number you must add to 135 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 2
38. 38.What is the remainder when 579 is divided by 11? (a) 4 (b) 8 (c) 6 (d) 7
39. 39.What is the units digit of 2^34? (a) 4 (b) 2 (c) 6 (d) 8
40. 40.What is the units digit of 4^53? (a) 6 (b) 3 (c) 4 (d) 5

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