Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 140 · 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 489 is divided by 8? (a) 0 (b) 2 (c) 7 (d) 1
2. 2.What is the units digit of 2^24? (a) 4 (b) 8 (c) 2 (d) 6
3. 3.Which of these numbers is divisible by 8? (a) 804 (b) 822 (c) 811 (d) 808
4. 4.What is the remainder when 412 is divided by 7? (a) 6 (b) 1 (c) 5 (d) 4
5. 5.What is the smallest number you must add to 221 to make it divisible by 3? (a) 1 (b) 2
6. 6.What is the units digit of 3^10? (a) 3 (b) 7 (c) 9 (d) 1
7. 7.What is the units digit of 3^13? (a) 7 (b) 1 (c) 9 (d) 3
8. 8.What is the remainder when 468 is divided by 11? (a) 5 (b) 8 (c) 6 (d) 7
9. 9.What is the remainder when 209 is divided by 3? (a) 1 (b) 2 (c) 0
10. 10.What is the units digit of 8^15? (a) 8 (b) 6 (c) 2 (d) 4
11. 11.What is the remainder when 90 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
12. 12.Which of these numbers is divisible by 11? (a) 1125 (b) 1134 (c) 1121 (d) 1122
13. 13.Which of these numbers is divisible by 8? (a) 3435 (b) 3454 (c) 3442 (d) 3440
14. 14.Which of these numbers is divisible by 6? (a) 2088 (b) 2093 (c) 2095 (d) 2087
15. 15.What is the units digit of 3^47? (a) 7 (b) 3 (c) 1 (d) 9
16. 16.Which of these numbers is divisible by 3? (a) 1859 (b) 1857 (c) 1861 (d) 1856
17. 17.What is the smallest number you must add to 503 to make it divisible by 7? (a) 1 (b) 3 (c) 6 (d) 2
18. 18.What is the remainder when 62 is divided by 3? (a) 1 (b) 2 (c) 0
19. 19.What is the remainder when 244 is divided by 9? (a) 2 (b) 1 (c) 8 (d) 0
20. 20.Which of these numbers is divisible by 4? (a) 554 (b) 562 (c) 556 (d) 559
21. 21.What is the remainder when 407 is divided by 7? (a) 0 (b) 2 (c) 6 (d) 1
22. 22.What is the smallest number you must add to 184 to make it divisible by 12? (a) 8 (b) 7 (c) 9 (d) 4
23. 23.Which of these numbers is divisible by 9? (a) 7951 (b) 7956 (c) 7957 (d) 7967
24. 24.Which of these numbers is divisible by 6? (a) 2988 (b) 2999 (c) 2985 (d) 2992
25. 25.What is the remainder when 104 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
26. 26.What is the units digit of 3^23? (a) 9 (b) 1 (c) 3 (d) 7
27. 27.Which of these numbers is divisible by 11? (a) 9553 (b) 9560 (c) 9543 (d) 9548
28. 28.What is the smallest number you must add to 291 to make it divisible by 4? (a) 2 (b) 3 (c) 1
29. 29.What is the units digit of 2^33? (a) 6 (b) 8 (c) 4 (d) 2
30. 30.What is the units digit of 7^11? (a) 1 (b) 9 (c) 7 (d) 3
31. 31.What is the remainder when 373 is divided by 9? (a) 3 (b) 5 (c) 6 (d) 4
32. 32.What is the units digit of 9^19? (a) 1 (b) 8 (c) 9 (d) 7
33. 33.Which of these numbers is divisible by 3? (a) 1022 (b) 1019 (c) 1020 (d) 1024
34. 34.What is the units digit of 7^18? (a) 9 (b) 3 (c) 7 (d) 1
35. 35.What is the remainder when 518 is divided by 8? (a) 5 (b) 2 (c) 6 (d) 7
36. 36.What is the remainder when 321 is divided by 5? (a) 2 (b) 0 (c) 4 (d) 1
37. 37.What is the remainder when 814 is divided by 12? (a) 11 (b) 2 (c) 10 (d) 9
38. 38.What is the units digit of 3^24? (a) 3 (b) 7 (c) 9 (d) 1
39. 39.What is the remainder when 273 is divided by 11? (a) 8 (b) 2 (c) 9 (d) 10
40. 40.What is the smallest number you must add to 417 to make it divisible by 10? (a) 7 (b) 4 (c) 2 (d) 3

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