Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 287 · 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 323 is divided by 6? (a) 1 (b) 5 (c) 3 (d) 4
2. 2.What is the units digit of 4^36? (a) 6 (b) 4 (c) 7 (d) 5
3. 3.What is the units digit of 8^12? (a) 6 (b) 8 (c) 2 (d) 4
4. 4.Which of these numbers is divisible by 3? (a) 238 (b) 241 (c) 244 (d) 240
5. 5.What is the smallest number you must add to 489 to make it divisible by 12? (a) 3 (b) 2 (c) 9 (d) 4
6. 6.What is the remainder when 62 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
7. 7.What is the smallest number you must add to 282 to make it divisible by 9? (a) 6 (b) 3 (c) 5 (d) 7
8. 8.Which of these numbers is divisible by 4? (a) 914 (b) 911 (c) 906 (d) 908
9. 9.What is the units digit of 9^51? (a) 9 (b) 7 (c) 1 (d) 8
10. 10.What is the remainder when 161 is divided by 6? (a) 3 (b) 5 (c) 4 (d) 1
11. 11.Which of these numbers is divisible by 6? (a) 821 (b) 812 (c) 805 (d) 810
12. 12.What is the remainder when 176 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 1
13. 13.What is the remainder when 168 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
14. 14.Which of these numbers is divisible by 9? (a) 4954 (b) 4961 (c) 4949 (d) 4950
15. 15.What is the units digit of 2^56? (a) 8 (b) 4 (c) 6 (d) 2
16. 16.Which of these numbers is divisible by 9? (a) 3308 (b) 3303 (c) 3315 (d) 3301
17. 17.Which of these numbers is divisible by 6? (a) 381 (b) 384 (c) 393 (d) 388
18. 18.What is the remainder when 706 is divided by 9? (a) 6 (b) 5 (c) 4 (d) 3
19. 19.What is the remainder when 82 is divided by 3? (a) 1 (b) 0 (c) 2
20. 20.What is the remainder when 169 is divided by 3? (a) 0 (b) 2 (c) 1
21. 21.Which of these numbers is divisible by 6? (a) 1117 (b) 1116 (c) 1125 (d) 1111
22. 22.What is the smallest number you must add to 592 to make it divisible by 9? (a) 1 (b) 3 (c) 7 (d) 2
23. 23.What is the units digit of 2^26? (a) 2 (b) 6 (c) 4 (d) 8
24. 24.What is the smallest number you must add to 678 to make it divisible by 9? (a) 3 (b) 7 (c) 6 (d) 5
25. 25.Which of these numbers is divisible by 11? (a) 9886 (b) 9867 (c) 9877 (d) 9857
26. 26.What is the remainder when 178 is divided by 12? (a) 9 (b) 11 (c) 2 (d) 10
27. 27.What is the remainder when 310 is divided by 9? (a) 4 (b) 3 (c) 6 (d) 5
28. 28.Which of these numbers is divisible by 9? (a) 648 (b) 661 (c) 649 (d) 647
29. 29.What is the units digit of 7^21? (a) 1 (b) 9 (c) 7 (d) 3
30. 30.Which of these numbers is divisible by 9? (a) 2268 (b) 2270 (c) 2266 (d) 2285
31. 31.What is the units digit of 3^19? (a) 9 (b) 7 (c) 3 (d) 1
32. 32.What is the smallest number you must add to 271 to make it divisible by 12? (a) 5 (b) 7 (c) 6 (d) 4
33. 33.What is the remainder when 449 is divided by 6? (a) 5 (b) 3 (c) 4 (d) 1
34. 34.What is the smallest number you must add to 134 to make it divisible by 3? (a) 2 (b) 1
35. 35.What is the units digit of 4^24? (a) 6 (b) 7 (c) 5 (d) 4
36. 36.What is the units digit of 2^19? (a) 4 (b) 6 (c) 2 (d) 8
37. 37.What is the smallest number you must add to 302 to make it divisible by 7? (a) 1 (b) 4 (c) 5 (d) 6
38. 38.What is the units digit of 2^48? (a) 4 (b) 6 (c) 8 (d) 2
39. 39.What is the units digit of 2^37? (a) 6 (b) 2 (c) 4 (d) 8
40. 40.What is the smallest number you must add to 62 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2

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