Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 489 · 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 423 is divided by 7? (a) 2 (b) 3 (c) 5 (d) 4
2. 2.What is the smallest number you must add to 112 to make it divisible by 3? (a) 1 (b) 2
3. 3.Which of these numbers is divisible by 8? (a) 3910 (b) 3896 (c) 3892 (d) 3899
4. 4.What is the smallest number you must add to 307 to make it divisible by 4? (a) 3 (b) 2 (c) 1
5. 5.Which of these numbers is divisible by 9? (a) 7192 (b) 7182 (c) 7190 (d) 7179
6. 6.What is the units digit of 4^49? (a) 3 (b) 4 (c) 6 (d) 5
7. 7.Which of these numbers is divisible by 6? (a) 3301 (b) 3295 (c) 3289 (d) 3294
8. 8.Which of these numbers is divisible by 3? (a) 1381 (b) 1375 (c) 1377 (d) 1379
9. 9.What is the smallest number you must add to 292 to make it divisible by 11? (a) 4 (b) 5 (c) 7 (d) 6
10. 10.What is the remainder when 445 is divided by 10? (a) 4 (b) 5 (c) 7 (d) 6
11. 11.What is the units digit of 7^56? (a) 7 (b) 3 (c) 1 (d) 9
12. 12.What is the units digit of 2^22? (a) 2 (b) 6 (c) 4 (d) 8
13. 13.What is the smallest number you must add to 235 to make it divisible by 7? (a) 5 (b) 4 (c) 2 (d) 3
14. 14.What is the remainder when 42 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
15. 15.What is the smallest number you must add to 378 to make it divisible by 8? (a) 7 (b) 6 (c) 2 (d) 5
16. 16.What is the remainder when 113 is divided by 11? (a) 8 (b) 3 (c) 4 (d) 2
17. 17.What is the remainder when 295 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
18. 18.What is the units digit of 2^30? (a) 6 (b) 4 (c) 2 (d) 8
19. 19.What is the units digit of 9^57? (a) 1 (b) 9 (c) 7 (d) 8
20. 20.Which of these numbers is divisible by 3? (a) 1439 (b) 1435 (c) 1434 (d) 1433
21. 21.What is the units digit of 7^13? (a) 3 (b) 1 (c) 9 (d) 7
22. 22.Which of these numbers is divisible by 8? (a) 3645 (b) 3662 (c) 3648 (d) 3654
23. 23.What is the units digit of 8^53? (a) 6 (b) 2 (c) 8 (d) 4
24. 24.What is the units digit of 9^33? (a) 9 (b) 8 (c) 1 (d) 7
25. 25.Which of these numbers is divisible by 11? (a) 3278 (b) 3281 (c) 3290 (d) 3271
26. 26.What is the remainder when 145 is divided by 3? (a) 1 (b) 2 (c) 0
27. 27.What is the units digit of 4^37? (a) 3 (b) 5 (c) 6 (d) 4
28. 28.Which of these numbers is divisible by 3? (a) 885 (b) 883 (c) 889 (d) 887
29. 29.What is the smallest number you must add to 501 to make it divisible by 11? (a) 7 (b) 4 (c) 6 (d) 5
30. 30.What is the smallest number you must add to 191 to make it divisible by 3? (a) 1 (b) 2
31. 31.What is the remainder when 176 is divided by 3? (a) 0 (b) 2 (c) 1
32. 32.Which of these numbers is divisible by 3? (a) 260 (b) 258 (c) 263 (d) 257
33. 33.What is the units digit of 8^54? (a) 8 (b) 2 (c) 4 (d) 6
34. 34.Which of these numbers is divisible by 6? (a) 3788 (b) 3795 (c) 3800 (d) 3792
35. 35.Which of these numbers is divisible by 6? (a) 755 (b) 759 (c) 756 (d) 764
36. 36.Which of these numbers is divisible by 8? (a) 597 (b) 605 (c) 592 (d) 585
37. 37.What is the smallest number you must add to 149 to make it divisible by 10? (a) 9 (b) 3 (c) 1 (d) 2
38. 38.What is the smallest number you must add to 547 to make it divisible by 7? (a) 5 (b) 6 (c) 1 (d) 4
39. 39.What is the remainder when 393 is divided by 10? (a) 7 (b) 4 (c) 3 (d) 2
40. 40.What is the smallest number you must add to 94 to make it divisible by 9? (a) 6 (b) 4 (c) 7 (d) 5

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