Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 526 · 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 9? (a) 7773 (b) 7780 (c) 7767 (d) 7760
2. 2.What is the units digit of 9^17? (a) 7 (b) 1 (c) 9 (d) 8
3. 3.Which of these numbers is divisible by 3? (a) 271 (b) 274 (c) 273 (d) 277
4. 4.Which of these numbers is divisible by 6? (a) 4679 (b) 4674 (c) 4673 (d) 4682
5. 5.Which of these numbers is divisible by 6? (a) 2984 (b) 2990 (c) 2997 (d) 2988
6. 6.What is the units digit of 8^41? (a) 2 (b) 8 (c) 6 (d) 4
7. 7.What is the remainder when 54 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
8. 8.What is the smallest number you must add to 49 to make it divisible by 4? (a) 2 (b) 1 (c) 3
9. 9.What is the remainder when 184 is divided by 6? (a) 5 (b) 3 (c) 4 (d) 2
10. 10.What is the units digit of 8^59? (a) 8 (b) 4 (c) 2 (d) 6
11. 11.What is the smallest number you must add to 301 to make it divisible by 4? (a) 2 (b) 1 (c) 3
12. 12.What is the units digit of 9^44? (a) 9 (b) 2 (c) 0 (d) 1
13. 13.What is the remainder when 182 is divided by 6? (a) 4 (b) 3 (c) 1 (d) 2
14. 14.What is the units digit of 8^47? (a) 6 (b) 8 (c) 4 (d) 2
15. 15.What is the smallest number you must add to 213 to make it divisible by 11? (a) 4 (b) 8 (c) 6 (d) 7
16. 16.Which of these numbers is divisible by 3? (a) 387 (b) 388 (c) 386 (d) 391
17. 17.Which of these numbers is divisible by 8? (a) 6486 (b) 6498 (c) 6488 (d) 6493
18. 18.What is the units digit of 2^43? (a) 4 (b) 2 (c) 8 (d) 6
19. 19.Which of these numbers is divisible by 3? (a) 1139 (b) 1136 (c) 1141 (d) 1137
20. 20.What is the smallest number you must add to 564 to make it divisible by 11? (a) 8 (b) 3 (c) 9 (d) 7
21. 21.What is the units digit of 4^53? (a) 5 (b) 4 (c) 6 (d) 3
22. 22.What is the remainder when 710 is divided by 11? (a) 7 (b) 8 (c) 6 (d) 5
23. 23.Which of these numbers is divisible by 4? (a) 3406 (b) 3409 (c) 3404 (d) 3401
24. 24.What is the units digit of 4^48? (a) 6 (b) 4 (c) 7 (d) 5
25. 25.What is the units digit of 2^47? (a) 6 (b) 4 (c) 8 (d) 2
26. 26.What is the units digit of 7^44? (a) 9 (b) 1 (c) 3 (d) 7
27. 27.What is the units digit of 8^44? (a) 2 (b) 8 (c) 6 (d) 4
28. 28.What is the smallest number you must add to 127 to make it divisible by 8? (a) 2 (b) 1 (c) 3 (d) 7
29. 29.What is the units digit of 9^33? (a) 8 (b) 9 (c) 1 (d) 7
30. 30.What is the remainder when 198 is divided by 7? (a) 5 (b) 2 (c) 1 (d) 3
31. 31.What is the smallest number you must add to 123 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
32. 32.What is the smallest number you must add to 190 to make it divisible by 8? (a) 6 (b) 3 (c) 2 (d) 1
33. 33.What is the units digit of 7^22? (a) 3 (b) 9 (c) 1 (d) 7
34. 34.What is the units digit of 3^33? (a) 9 (b) 3 (c) 7 (d) 1
35. 35.What is the remainder when 37 is divided by 3? (a) 2 (b) 0 (c) 1
36. 36.Which of these numbers is divisible by 8? (a) 6833 (b) 6824 (c) 6827 (d) 6817
37. 37.What is the remainder when 246 is divided by 5? (a) 0 (b) 2 (c) 1 (d) 4
38. 38.What is the remainder when 292 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
39. 39.What is the remainder when 85 is divided by 8? (a) 3 (b) 4 (c) 6 (d) 5
40. 40.What is the remainder when 394 is divided by 9? (a) 7 (b) 2 (c) 8 (d) 6

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