Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 821 · 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 4? (a) 2302 (b) 2307 (c) 2300 (d) 2298
2. 2.Which of these numbers is divisible by 9? (a) 1193 (b) 1184 (c) 1203 (d) 1188
3. 3.What is the remainder when 177 is divided by 6? (a) 4 (b) 3 (c) 5 (d) 2
4. 4.What is the units digit of 9^13? (a) 7 (b) 9 (c) 8 (d) 1
5. 5.What is the smallest number you must add to 223 to make it divisible by 4? (a) 2 (b) 1 (c) 3
6. 6.What is the remainder when 145 is divided by 12? (a) 11 (b) 0 (c) 2 (d) 1
7. 7.What is the remainder when 502 is divided by 9? (a) 7 (b) 2 (c) 6 (d) 8
8. 8.Which of these numbers is divisible by 11? (a) 1430 (b) 1428 (c) 1438 (d) 1442
9. 9.What is the remainder when 649 is divided by 9? (a) 1 (b) 8 (c) 2 (d) 0
10. 10.What is the remainder when 249 is divided by 10? (a) 7 (b) 1 (c) 9 (d) 8
11. 11.What is the units digit of 4^29? (a) 4 (b) 5 (c) 3 (d) 6
12. 12.What is the remainder when 453 is divided by 10? (a) 2 (b) 7 (c) 4 (d) 3
13. 13.What is the units digit of 3^27? (a) 9 (b) 1 (c) 3 (d) 7
14. 14.What is the smallest number you must add to 93 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
15. 15.What is the remainder when 528 is divided by 10? (a) 2 (b) 8 (c) 7 (d) 9
16. 16.What is the remainder when 95 is divided by 6? (a) 1 (b) 4 (c) 5 (d) 3
17. 17.What is the smallest number you must add to 435 to make it divisible by 8? (a) 6 (b) 4 (c) 3 (d) 5
18. 18.Which of these numbers is divisible by 8? (a) 3762 (b) 3756 (c) 3760 (d) 3770
19. 19.What is the remainder when 208 is divided by 7? (a) 6 (b) 2 (c) 4 (d) 5
20. 20.What is the remainder when 577 is divided by 9? (a) 8 (b) 0 (c) 2 (d) 1
21. 21.What is the remainder when 434 is divided by 12? (a) 2 (b) 1 (c) 3 (d) 10
22. 22.Which of these numbers is divisible by 9? (a) 6381 (b) 6373 (c) 6393 (d) 6383
23. 23.What is the units digit of 8^22? (a) 6 (b) 4 (c) 8 (d) 2
24. 24.What is the remainder when 322 is divided by 8? (a) 6 (b) 1 (c) 2 (d) 3
25. 25.Which of these numbers is divisible by 9? (a) 4796 (b) 4808 (c) 4800 (d) 4797
26. 26.Which of these numbers is divisible by 11? (a) 6467 (b) 6440 (c) 6446 (d) 6449
27. 27.What is the smallest number you must add to 168 to make it divisible by 11? (a) 8 (b) 9 (c) 7 (d) 3
28. 28.What is the units digit of 8^27? (a) 4 (b) 8 (c) 6 (d) 2
29. 29.What is the remainder when 119 is divided by 3? (a) 0 (b) 2 (c) 1
30. 30.What is the units digit of 8^38? (a) 8 (b) 2 (c) 6 (d) 4
31. 31.What is the units digit of 7^24? (a) 7 (b) 1 (c) 3 (d) 9
32. 32.What is the units digit of 7^33? (a) 9 (b) 3 (c) 7 (d) 1
33. 33.What is the units digit of 9^41? (a) 1 (b) 7 (c) 9 (d) 8
34. 34.What is the remainder when 67 is divided by 3? (a) 2 (b) 1 (c) 0
35. 35.What is the smallest number you must add to 185 to make it divisible by 4? (a) 1 (b) 2 (c) 3
36. 36.What is the units digit of 4^53? (a) 3 (b) 4 (c) 6 (d) 5
37. 37.What is the smallest number you must add to 400 to make it divisible by 7? (a) 1 (b) 4 (c) 6 (d) 5
38. 38.What is the units digit of 9^44? (a) 1 (b) 2 (c) 9 (d) 0
39. 39.What is the remainder when 93 is divided by 6? (a) 4 (b) 2 (c) 3 (d) 5
40. 40.What is the remainder when 105 is divided by 10? (a) 4 (b) 5 (c) 6 (d) 7

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