Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 777 · 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 smallest number you must add to 705 to make it divisible by 11? (a) 10 (b) 1 (c) 9 (d) 8
2. 2.Which of these numbers is divisible by 3? (a) 2293 (b) 2296 (c) 2292 (d) 2291
3. 3.Which of these numbers is divisible by 9? (a) 3659 (b) 3648 (c) 3664 (d) 3654
4. 4.Which of these numbers is divisible by 4? (a) 1602 (b) 1597 (c) 1605 (d) 1600
5. 5.Which of these numbers is divisible by 4? (a) 3326 (b) 3333 (c) 3329 (d) 3328
6. 6.What is the units digit of 9^12? (a) 9 (b) 1 (c) 2 (d) 0
7. 7.What is the smallest number you must add to 822 to make it divisible by 12? (a) 8 (b) 7 (c) 5 (d) 6
8. 8.What is the units digit of 3^28? (a) 7 (b) 1 (c) 3 (d) 9
9. 9.What is the smallest number you must add to 178 to make it divisible by 7? (a) 4 (b) 6 (c) 3 (d) 5
10. 10.What is the remainder when 305 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
11. 11.What is the units digit of 9^58? (a) 2 (b) 0 (c) 1 (d) 9
12. 12.What is the units digit of 8^42? (a) 6 (b) 4 (c) 2 (d) 8
13. 13.What is the units digit of 7^28? (a) 9 (b) 3 (c) 1 (d) 7
14. 14.What is the remainder when 233 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
15. 15.What is the smallest number you must add to 132 to make it divisible by 9? (a) 4 (b) 2 (c) 6 (d) 3
16. 16.Which of these numbers is divisible by 4? (a) 395 (b) 392 (c) 397 (d) 389
17. 17.What is the remainder when 294 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
18. 18.What is the units digit of 2^13? (a) 2 (b) 4 (c) 8 (d) 6
19. 19.Which of these numbers is divisible by 11? (a) 1496 (b) 1498 (c) 1491 (d) 1513
20. 20.Which of these numbers is divisible by 3? (a) 1969 (b) 1976 (c) 1973 (d) 1971
21. 21.What is the smallest number you must add to 46 to make it divisible by 3? (a) 1 (b) 2
22. 22.What is the units digit of 4^28? (a) 5 (b) 6 (c) 4 (d) 7
23. 23.What is the units digit of 7^59? (a) 7 (b) 3 (c) 9 (d) 1
24. 24.What is the units digit of 7^30? (a) 7 (b) 3 (c) 1 (d) 9
25. 25.Which of these numbers is divisible by 11? (a) 1352 (b) 1360 (c) 1365 (d) 1353
26. 26.What is the remainder when 274 is divided by 5? (a) 3 (b) 4 (c) 2 (d) 1
27. 27.What is the remainder when 230 is divided by 11? (a) 1 (b) 8 (c) 9 (d) 10
28. 28.Which of these numbers is divisible by 11? (a) 9313 (b) 9306 (c) 9318 (d) 9305
29. 29.Which of these numbers is divisible by 11? (a) 3371 (b) 3381 (c) 3377 (d) 3398
30. 30.What is the smallest number you must add to 281 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
31. 31.What is the remainder when 401 is divided by 9? (a) 4 (b) 7 (c) 5 (d) 6
32. 32.What is the remainder when 966 is divided by 12? (a) 5 (b) 7 (c) 8 (d) 6
33. 33.Which of these numbers is divisible by 4? (a) 3506 (b) 3501 (c) 3500 (d) 3498
34. 34.What is the remainder when 626 is divided by 8? (a) 6 (b) 3 (c) 1 (d) 2
35. 35.What is the units digit of 4^20? (a) 6 (b) 7 (c) 4 (d) 5
36. 36.What is the units digit of 2^24? (a) 6 (b) 2 (c) 8 (d) 4
37. 37.What is the smallest number you must add to 137 to make it divisible by 7? (a) 2 (b) 4 (c) 3 (d) 5
38. 38.Which of these numbers is divisible by 4? (a) 461 (b) 459 (c) 467 (d) 460
39. 39.What is the remainder when 518 is divided by 8? (a) 5 (b) 7 (c) 2 (d) 6
40. 40.What is the smallest number you must add to 434 to make it divisible by 8? (a) 5 (b) 2 (c) 7 (d) 6

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