Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 869 · 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) 711 (b) 714 (c) 716 (d) 722
2. 2.What is the remainder when 323 is divided by 9? (a) 1 (b) 6 (c) 7 (d) 8
3. 3.What is the units digit of 4^27? (a) 6 (b) 3 (c) 4 (d) 5
4. 4.Which of these numbers is divisible by 6? (a) 1451 (b) 1446 (c) 1457 (d) 1441
5. 5.What is the remainder when 533 is divided by 10? (a) 7 (b) 4 (c) 3 (d) 2
6. 6.Which of these numbers is divisible by 9? (a) 5119 (b) 5112 (c) 5108 (d) 5127
7. 7.What is the units digit of 3^20? (a) 7 (b) 3 (c) 1 (d) 9
8. 8.Which of these numbers is divisible by 9? (a) 5293 (b) 5287 (c) 5280 (d) 5283
9. 9.What is the remainder when 615 is divided by 9? (a) 2 (b) 3 (c) 6 (d) 4
10. 10.What is the units digit of 9^26? (a) 9 (b) 2 (c) 1 (d) 0
11. 11.What is the remainder when 310 is divided by 12? (a) 10 (b) 11 (c) 2 (d) 9
12. 12.Which of these numbers is divisible by 6? (a) 1509 (b) 1506 (c) 1502 (d) 1514
13. 13.Which of these numbers is divisible by 8? (a) 3145 (b) 3129 (c) 3136 (d) 3143
14. 14.What is the smallest number you must add to 363 to make it divisible by 8? (a) 3 (b) 6 (c) 4 (d) 5
15. 15.What is the units digit of 7^15? (a) 1 (b) 3 (c) 7 (d) 9
16. 16.What is the smallest number you must add to 579 to make it divisible by 11? (a) 7 (b) 5 (c) 3 (d) 4
17. 17.What is the smallest number you must add to 53 to make it divisible by 4? (a) 2 (b) 1 (c) 3
18. 18.What is the units digit of 7^24? (a) 9 (b) 1 (c) 7 (d) 3
19. 19.What is the units digit of 8^54? (a) 2 (b) 8 (c) 4 (d) 6
20. 20.What is the units digit of 8^59? (a) 2 (b) 6 (c) 4 (d) 8
21. 21.What is the remainder when 378 is divided by 11? (a) 7 (b) 3 (c) 4 (d) 5
22. 22.What is the units digit of 9^45? (a) 1 (b) 8 (c) 9 (d) 7
23. 23.Which of these numbers is divisible by 4? (a) 2670 (b) 2668 (c) 2666 (d) 2674
24. 24.What is the remainder when 382 is divided by 9? (a) 3 (b) 6 (c) 4 (d) 5
25. 25.What is the remainder when 181 is divided by 5? (a) 4 (b) 1 (c) 0 (d) 2
26. 26.What is the units digit of 4^57? (a) 5 (b) 6 (c) 3 (d) 4
27. 27.Which of these numbers is divisible by 8? (a) 3058 (b) 3073 (c) 3068 (d) 3064
28. 28.What is the remainder when 777 is divided by 12? (a) 10 (b) 3 (c) 9 (d) 8
29. 29.Which of these numbers is divisible by 4? (a) 1461 (b) 1467 (c) 1464 (d) 1470
30. 30.What is the smallest number you must add to 530 to make it divisible by 12? (a) 11 (b) 10 (c) 2 (d) 9
31. 31.What is the units digit of 3^42? (a) 1 (b) 3 (c) 7 (d) 9
32. 32.What is the smallest number you must add to 129 to make it divisible by 6? (a) 2 (b) 4 (c) 5 (d) 3
33. 33.What is the remainder when 175 is divided by 3? (a) 2 (b) 0 (c) 1
34. 34.Which of these numbers is divisible by 11? (a) 1254 (b) 1255 (c) 1268 (d) 1253
35. 35.Which of these numbers is divisible by 3? (a) 2320 (b) 2326 (c) 2322 (d) 2323
36. 36.Which of these numbers is divisible by 11? (a) 6563 (b) 6572 (c) 6556 (d) 6555
37. 37.What is the units digit of 9^46? (a) 9 (b) 0 (c) 2 (d) 1
38. 38.What is the smallest number you must add to 696 to make it divisible by 10? (a) 6 (b) 4 (c) 3 (d) 5
39. 39.What is the units digit of 8^44? (a) 8 (b) 2 (c) 6 (d) 4
40. 40.What is the smallest number you must add to 456 to make it divisible by 7? (a) 4 (b) 1 (c) 6 (d) 5

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