Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 357 · 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.Which of these numbers is divisible by 3? (a) 2341 (b) 2339 (c) 2344 (d) 2340
2. 2.Which of these numbers is divisible by 9? (a) 2326 (b) 2338 (c) 2345 (d) 2331
3. 3.Which of these numbers is divisible by 6? (a) 1904 (b) 1910 (c) 1908 (d) 1916
4. 4.What is the units digit of 7^40? (a) 7 (b) 9 (c) 3 (d) 1
5. 5.What is the smallest number you must add to 577 to make it divisible by 9? (a) 7 (b) 8 (c) 1 (d) 6
6. 6.What is the units digit of 3^15? (a) 9 (b) 7 (c) 1 (d) 3
7. 7.What is the units digit of 3^33? (a) 9 (b) 3 (c) 7 (d) 1
8. 8.What is the smallest number you must add to 99 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 2
9. 9.Which of these numbers is divisible by 11? (a) 2731 (b) 2749 (c) 2718 (d) 2728
10. 10.What is the remainder when 185 is divided by 10? (a) 4 (b) 7 (c) 6 (d) 5
11. 11.What is the smallest number you must add to 141 to make it divisible by 7? (a) 4 (b) 5 (c) 1 (d) 6
12. 12.What is the remainder when 483 is divided by 10? (a) 2 (b) 4 (c) 7 (d) 3
13. 13.Which of these numbers is divisible by 8? (a) 5792 (b) 5802 (c) 5798 (d) 5789
14. 14.What is the units digit of 8^42? (a) 2 (b) 8 (c) 4 (d) 6
15. 15.What is the units digit of 8^43? (a) 4 (b) 2 (c) 8 (d) 6
16. 16.Which of these numbers is divisible by 9? (a) 5975 (b) 5976 (c) 5987 (d) 5977
17. 17.What is the smallest number you must add to 146 to make it divisible by 4? (a) 2 (b) 1 (c) 3
18. 18.What is the smallest number you must add to 88 to make it divisible by 3? (a) 1 (b) 2
19. 19.What is the remainder when 208 is divided by 7? (a) 2 (b) 4 (c) 5 (d) 6
20. 20.Which of these numbers is divisible by 4? (a) 207 (b) 203 (c) 200 (d) 199
21. 21.Which of these numbers is divisible by 4? (a) 2875 (b) 2879 (c) 2876 (d) 2882
22. 22.Which of these numbers is divisible by 6? (a) 4831 (b) 4838 (c) 4836 (d) 4847
23. 23.What is the remainder when 184 is divided by 12? (a) 5 (b) 3 (c) 8 (d) 4
24. 24.Which of these numbers is divisible by 9? (a) 3167 (b) 3150 (c) 3151 (d) 3145
25. 25.What is the remainder when 429 is divided by 7? (a) 1 (b) 5 (c) 3 (d) 2
26. 26.What is the units digit of 4^38? (a) 4 (b) 6 (c) 5 (d) 7
27. 27.What is the units digit of 7^34? (a) 1 (b) 7 (c) 9 (d) 3
28. 28.Which of these numbers is divisible by 4? (a) 1711 (b) 1719 (c) 1714 (d) 1712
29. 29.What is the remainder when 307 is divided by 9? (a) 2 (b) 0 (c) 8 (d) 1
30. 30.What is the smallest number you must add to 370 to make it divisible by 8? (a) 7 (b) 5 (c) 6 (d) 2
31. 31.What is the units digit of 4^24? (a) 4 (b) 7 (c) 6 (d) 5
32. 32.What is the remainder when 42 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
33. 33.Which of these numbers is divisible by 6? (a) 4170 (b) 4181 (c) 4171 (d) 4169
34. 34.What is the smallest number you must add to 113 to make it divisible by 11? (a) 7 (b) 3 (c) 8 (d) 9
35. 35.Which of these numbers is divisible by 6? (a) 560 (b) 565 (c) 572 (d) 564
36. 36.What is the smallest number you must add to 77 to make it divisible by 6? (a) 3 (b) 5 (c) 2 (d) 1
37. 37.What is the smallest number you must add to 100 to make it divisible by 3? (a) 1 (b) 2
38. 38.Which of these numbers is divisible by 9? (a) 6960 (b) 6956 (c) 6970 (d) 6957
39. 39.What is the remainder when 262 is divided by 10? (a) 2 (b) 3 (c) 1 (d) 8
40. 40.What is the units digit of 3^47? (a) 3 (b) 9 (c) 1 (d) 7

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