Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 159 · 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 6? (a) 449 (b) 450 (c) 457 (d) 455
2. 2.What is the smallest number you must add to 467 to make it divisible by 8? (a) 6 (b) 4 (c) 3 (d) 5
3. 3.What is the smallest number you must add to 158 to make it divisible by 6? (a) 3 (b) 4 (c) 2 (d) 5
4. 4.What is the units digit of 2^56? (a) 4 (b) 2 (c) 6 (d) 8
5. 5.What is the units digit of 3^36? (a) 1 (b) 3 (c) 7 (d) 9
6. 6.What is the units digit of 4^27? (a) 5 (b) 6 (c) 4 (d) 3
7. 7.What is the remainder when 381 is divided by 8? (a) 3 (b) 5 (c) 4 (d) 6
8. 8.Which of these numbers is divisible by 6? (a) 4990 (b) 4996 (c) 4983 (d) 4986
9. 9.What is the units digit of 8^59? (a) 4 (b) 2 (c) 6 (d) 8
10. 10.Which of these numbers is divisible by 6? (a) 3743 (b) 3748 (c) 3733 (d) 3738
11. 11.Which of these numbers is divisible by 6? (a) 503 (b) 497 (c) 509 (d) 498
12. 12.What is the remainder when 431 is divided by 11? (a) 1 (b) 3 (c) 2 (d) 9
13. 13.Which of these numbers is divisible by 3? (a) 1343 (b) 1340 (c) 1337 (d) 1338
14. 14.What is the smallest number you must add to 178 to make it divisible by 3? (a) 1 (b) 2
15. 15.What is the units digit of 4^45? (a) 3 (b) 4 (c) 5 (d) 6
16. 16.Which of these numbers is divisible by 4? (a) 3350 (b) 3357 (c) 3355 (d) 3352
17. 17.What is the remainder when 224 is divided by 3? (a) 1 (b) 2 (c) 0
18. 18.What is the units digit of 9^54? (a) 0 (b) 2 (c) 1 (d) 9
19. 19.What is the units digit of 9^59? (a) 7 (b) 9 (c) 1 (d) 8
20. 20.What is the units digit of 9^19? (a) 9 (b) 1 (c) 7 (d) 8
21. 21.What is the remainder when 72 is divided by 7? (a) 2 (b) 3 (c) 1 (d) 5
22. 22.Which of these numbers is divisible by 6? (a) 3174 (b) 3182 (c) 3178 (d) 3172
23. 23.What is the units digit of 2^14? (a) 6 (b) 8 (c) 2 (d) 4
24. 24.What is the units digit of 7^22? (a) 3 (b) 9 (c) 1 (d) 7
25. 25.Which of these numbers is divisible by 8? (a) 7035 (b) 7053 (c) 7041 (d) 7040
26. 26.Which of these numbers is divisible by 9? (a) 5541 (b) 5559 (c) 5544 (d) 5548
27. 27.What is the remainder when 226 is divided by 11? (a) 7 (b) 8 (c) 6 (d) 5
28. 28.What is the units digit of 3^29? (a) 3 (b) 9 (c) 7 (d) 1
29. 29.What is the units digit of 7^39? (a) 7 (b) 9 (c) 3 (d) 1
30. 30.What is the units digit of 4^29? (a) 4 (b) 5 (c) 6 (d) 3
31. 31.Which of these numbers is divisible by 9? (a) 5062 (b) 5067 (c) 5075 (d) 5083
32. 32.What is the units digit of 2^31? (a) 6 (b) 2 (c) 4 (d) 8
33. 33.Which of these numbers is divisible by 4? (a) 3050 (b) 3053 (c) 3046 (d) 3048
34. 34.What is the smallest number you must add to 185 to make it divisible by 12? (a) 6 (b) 7 (c) 5 (d) 8
35. 35.Which of these numbers is divisible by 8? (a) 2512 (b) 2511 (c) 2518 (d) 2523
36. 36.What is the smallest number you must add to 331 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 5
37. 37.What is the smallest number you must add to 145 to make it divisible by 3? (a) 1 (b) 2
38. 38.What is the units digit of 7^30? (a) 9 (b) 1 (c) 3 (d) 7
39. 39.What is the units digit of 2^28? (a) 8 (b) 6 (c) 4 (d) 2
40. 40.What is the remainder when 662 is divided by 10? (a) 3 (b) 2 (c) 8 (d) 1

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