Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 443 · 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 remainder when 61 is divided by 3? (a) 1 (b) 2 (c) 0
2. 2.What is the smallest number you must add to 489 to make it divisible by 10? (a) 1 (b) 2 (c) 3 (d) 9
3. 3.What is the remainder when 226 is divided by 3? (a) 2 (b) 1 (c) 0
4. 4.What is the remainder when 213 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
5. 5.What is the remainder when 235 is divided by 12? (a) 7 (b) 5 (c) 8 (d) 6
6. 6.What is the units digit of 2^20? (a) 2 (b) 8 (c) 6 (d) 4
7. 7.Which of these numbers is divisible by 8? (a) 2223 (b) 2214 (c) 2208 (d) 2204
8. 8.What is the smallest number you must add to 326 to make it divisible by 9? (a) 6 (b) 7 (c) 2 (d) 8
9. 9.Which of these numbers is divisible by 11? (a) 9025 (b) 9033 (c) 9011 (d) 9020
10. 10.What is the smallest number you must add to 138 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
11. 11.What is the units digit of 9^47? (a) 8 (b) 9 (c) 1 (d) 7
12. 12.Which of these numbers is divisible by 8? (a) 3831 (b) 3832 (c) 3847 (d) 3834
13. 13.What is the smallest number you must add to 209 to make it divisible by 4? (a) 1 (b) 2 (c) 3
14. 14.What is the smallest number you must add to 307 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 1
15. 15.What is the smallest number you must add to 189 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
16. 16.Which of these numbers is divisible by 6? (a) 4134 (b) 4139 (c) 4133 (d) 4141
17. 17.What is the smallest number you must add to 295 to make it divisible by 8? (a) 1 (b) 3 (c) 2 (d) 7
18. 18.Which of these numbers is divisible by 4? (a) 3230 (b) 3225 (c) 3223 (d) 3224
19. 19.Which of these numbers is divisible by 3? (a) 1798 (b) 1795 (c) 1801 (d) 1797
20. 20.Which of these numbers is divisible by 6? (a) 255 (b) 246 (c) 250 (d) 243
21. 21.What is the remainder when 205 is divided by 3? (a) 0 (b) 1 (c) 2
22. 22.What is the remainder when 208 is divided by 6? (a) 3 (b) 4 (c) 5 (d) 2
23. 23.Which of these numbers is divisible by 11? (a) 3820 (b) 3815 (c) 3817 (d) 3831
24. 24.What is the remainder when 249 is divided by 6? (a) 5 (b) 3 (c) 4 (d) 2
25. 25.Which of these numbers is divisible by 9? (a) 2304 (b) 2302 (c) 2316 (d) 2309
26. 26.What is the smallest number you must add to 224 to make it divisible by 3? (a) 1 (b) 2
27. 27.Which of these numbers is divisible by 8? (a) 596 (b) 584 (c) 582 (d) 588
28. 28.Which of these numbers is divisible by 9? (a) 7194 (b) 7208 (c) 7214 (d) 7200
29. 29.What is the units digit of 2^10? (a) 8 (b) 2 (c) 4 (d) 6
30. 30.Which of these numbers is divisible by 4? (a) 1983 (b) 1984 (c) 1989 (d) 1985
31. 31.What is the units digit of 3^51? (a) 3 (b) 7 (c) 1 (d) 9
32. 32.Which of these numbers is divisible by 3? (a) 1913 (b) 1908 (c) 1909 (d) 1907
33. 33.What is the smallest number you must add to 243 to make it divisible by 7? (a) 5 (b) 3 (c) 2 (d) 1
34. 34.What is the smallest number you must add to 525 to make it divisible by 9? (a) 5 (b) 7 (c) 3 (d) 6
35. 35.Which of these numbers is divisible by 11? (a) 6235 (b) 6237 (c) 6239 (d) 6258
36. 36.Which of these numbers is divisible by 9? (a) 3529 (b) 3519 (c) 3511 (d) 3527
37. 37.Which of these numbers is divisible by 4? (a) 1905 (b) 1902 (c) 1904 (d) 1910
38. 38.What is the remainder when 264 is divided by 7? (a) 6 (b) 2 (c) 5 (d) 4
39. 39.Which of these numbers is divisible by 6? (a) 1797 (b) 1805 (c) 1794 (d) 1791
40. 40.Which of these numbers is divisible by 3? (a) 695 (b) 693 (c) 692 (d) 697

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