Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 904 · 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 smallest number you must add to 496 to make it divisible by 11? (a) 9 (b) 8 (c) 1 (d) 10
2. 2.Which of these numbers is divisible by 9? (a) 5482 (b) 5490 (c) 5507 (d) 5498
3. 3.What is the units digit of 4^49? (a) 5 (b) 6 (c) 4 (d) 3
4. 4.What is the remainder when 647 is divided by 8? (a) 6 (b) 1 (c) 7 (d) 5
5. 5.What is the units digit of 9^57? (a) 1 (b) 7 (c) 9 (d) 8
6. 6.What is the units digit of 4^30? (a) 7 (b) 4 (c) 5 (d) 6
7. 7.Which of these numbers is divisible by 3? (a) 1785 (b) 1789 (c) 1784 (d) 1787
8. 8.What is the units digit of 8^14? (a) 2 (b) 8 (c) 4 (d) 6
9. 9.Which of these numbers is divisible by 9? (a) 1025 (b) 1017 (c) 1028 (d) 1013
10. 10.What is the remainder when 397 is divided by 7? (a) 6 (b) 2 (c) 4 (d) 5
11. 11.What is the remainder when 365 is divided by 8? (a) 4 (b) 5 (c) 6 (d) 3
12. 12.What is the remainder when 263 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
13. 13.Which of these numbers is divisible by 9? (a) 7691 (b) 7697 (c) 7679 (d) 7686
14. 14.What is the units digit of 9^17? (a) 8 (b) 9 (c) 1 (d) 7
15. 15.What is the remainder when 144 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
16. 16.Which of these numbers is divisible by 4? (a) 2125 (b) 2121 (c) 2124 (d) 2130
17. 17.What is the units digit of 4^57? (a) 6 (b) 5 (c) 3 (d) 4
18. 18.What is the units digit of 4^41? (a) 6 (b) 4 (c) 5 (d) 3
19. 19.What is the remainder when 366 is divided by 7? (a) 3 (b) 1 (c) 5 (d) 2
20. 20.What is the smallest number you must add to 437 to make it divisible by 8? (a) 4 (b) 5 (c) 3 (d) 2
21. 21.What is the smallest number you must add to 160 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 2
22. 22.What is the units digit of 4^34? (a) 4 (b) 7 (c) 5 (d) 6
23. 23.What is the units digit of 9^10? (a) 2 (b) 1 (c) 9 (d) 0
24. 24.What is the units digit of 2^14? (a) 4 (b) 2 (c) 6 (d) 8
25. 25.What is the smallest number you must add to 226 to make it divisible by 7? (a) 5 (b) 4 (c) 6 (d) 2
26. 26.Which of these numbers is divisible by 3? (a) 1877 (b) 1872 (c) 1870 (d) 1874
27. 27.What is the smallest number you must add to 532 to make it divisible by 10? (a) 8 (b) 2 (c) 7 (d) 9
28. 28.Which of these numbers is divisible by 8? (a) 1369 (b) 1353 (c) 1364 (d) 1360
29. 29.What is the remainder when 299 is divided by 9? (a) 7 (b) 3 (c) 1 (d) 2
30. 30.Which of these numbers is divisible by 4? (a) 330 (b) 328 (c) 333 (d) 327
31. 31.What is the remainder when 236 is divided by 12? (a) 9 (b) 4 (c) 7 (d) 8
32. 32.What is the smallest number you must add to 143 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 1
33. 33.What is the units digit of 2^53? (a) 8 (b) 2 (c) 4 (d) 6
34. 34.What is the units digit of 3^49? (a) 9 (b) 1 (c) 7 (d) 3
35. 35.What is the remainder when 185 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
36. 36.What is the remainder when 250 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
37. 37.What is the smallest number you must add to 263 to make it divisible by 12? (a) 1 (b) 11 (c) 2 (d) 3
38. 38.What is the units digit of 4^40? (a) 6 (b) 7 (c) 4 (d) 5
39. 39.What is the remainder when 275 is divided by 9? (a) 5 (b) 6 (c) 7 (d) 4
40. 40.What is the remainder when 208 is divided by 7? (a) 5 (b) 4 (c) 2 (d) 6

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