Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 736 · 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 206 to make it divisible by 3? (a) 2 (b) 1
2. 2.Which of these numbers is divisible by 8? (a) 2936 (b) 2949 (c) 2930 (d) 2943
3. 3.Which of these numbers is divisible by 4? (a) 293 (b) 292 (c) 290 (d) 298
4. 4.Which of these numbers is divisible by 4? (a) 2121 (b) 2125 (c) 2120 (d) 2119
5. 5.What is the smallest number you must add to 465 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 3
6. 6.What is the units digit of 4^29? (a) 4 (b) 6 (c) 3 (d) 5
7. 7.Which of these numbers is divisible by 6? (a) 2652 (b) 2654 (c) 2650 (d) 2663
8. 8.What is the smallest number you must add to 459 to make it divisible by 10? (a) 1 (b) 9 (c) 3 (d) 2
9. 9.What is the smallest number you must add to 292 to make it divisible by 8? (a) 4 (b) 3 (c) 5 (d) 6
10. 10.What is the remainder when 551 is divided by 10? (a) 1 (b) 9 (c) 0 (d) 2
11. 11.Which of these numbers is divisible by 11? (a) 3130 (b) 3136 (c) 3116 (d) 3124
12. 12.What is the smallest number you must add to 595 to make it divisible by 10? (a) 5 (b) 6 (c) 4 (d) 7
13. 13.What is the units digit of 9^37? (a) 1 (b) 8 (c) 7 (d) 9
14. 14.What is the remainder when 374 is divided by 12? (a) 2 (b) 3 (c) 10 (d) 1
15. 15.What is the smallest number you must add to 343 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
16. 16.What is the units digit of 3^15? (a) 1 (b) 3 (c) 9 (d) 7
17. 17.Which of these numbers is divisible by 8? (a) 2245 (b) 2240 (c) 2234 (d) 2254
18. 18.What is the units digit of 4^40? (a) 4 (b) 5 (c) 7 (d) 6
19. 19.What is the units digit of 3^48? (a) 9 (b) 1 (c) 7 (d) 3
20. 20.What is the remainder when 530 is divided by 8? (a) 1 (b) 2 (c) 3 (d) 6
21. 21.What is the smallest number you must add to 169 to make it divisible by 4? (a) 1 (b) 3 (c) 2
22. 22.What is the remainder when 366 is divided by 7? (a) 3 (b) 2 (c) 5 (d) 1
23. 23.Which of these numbers is divisible by 6? (a) 4007 (b) 4002 (c) 4009 (d) 3999
24. 24.Which of these numbers is divisible by 6? (a) 2096 (b) 2089 (c) 2088 (d) 2083
25. 25.Which of these numbers is divisible by 6? (a) 1715 (b) 1704 (c) 1705 (d) 1702
26. 26.What is the remainder when 232 is divided by 7? (a) 0 (b) 2 (c) 1 (d) 6
27. 27.What is the units digit of 2^52? (a) 8 (b) 4 (c) 2 (d) 6
28. 28.What is the remainder when 162 is divided by 7? (a) 0 (b) 6 (c) 1 (d) 2
29. 29.What is the remainder when 295 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
30. 30.Which of these numbers is divisible by 4? (a) 888 (b) 889 (c) 894 (d) 887
31. 31.What is the smallest number you must add to 53 to make it divisible by 4? (a) 3 (b) 1 (c) 2
32. 32.What is the smallest number you must add to 762 to make it divisible by 12? (a) 8 (b) 7 (c) 6 (d) 5
33. 33.Which of these numbers is divisible by 3? (a) 2173 (b) 2177 (c) 2172 (d) 2170
34. 34.What is the remainder when 218 is divided by 9? (a) 2 (b) 3 (c) 7 (d) 1
35. 35.What is the smallest number you must add to 844 to make it divisible by 12? (a) 4 (b) 9 (c) 7 (d) 8
36. 36.What is the remainder when 91 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
37. 37.What is the units digit of 8^32? (a) 6 (b) 4 (c) 8 (d) 2
38. 38.What is the units digit of 8^42? (a) 2 (b) 4 (c) 8 (d) 6
39. 39.What is the units digit of 8^12? (a) 8 (b) 6 (c) 4 (d) 2
40. 40.Which of these numbers is divisible by 9? (a) 4226 (b) 4204 (c) 4220 (d) 4212

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