Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 149 · 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 146 to make it divisible by 3? (a) 1 (b) 2
2. 2.What is the remainder when 580 is divided by 12? (a) 5 (b) 3 (c) 4 (d) 8
3. 3.What is the smallest number you must add to 784 to make it divisible by 10? (a) 5 (b) 7 (c) 6 (d) 4
4. 4.What is the smallest number you must add to 226 to make it divisible by 3? (a) 2 (b) 1
5. 5.What is the smallest number you must add to 542 to make it divisible by 10? (a) 2 (b) 7 (c) 9 (d) 8
6. 6.What is the units digit of 4^55? (a) 5 (b) 4 (c) 3 (d) 6
7. 7.What is the units digit of 4^53? (a) 3 (b) 5 (c) 4 (d) 6
8. 8.Which of these numbers is divisible by 4? (a) 799 (b) 800 (c) 807 (d) 803
9. 9.Which of these numbers is divisible by 8? (a) 6208 (b) 6205 (c) 6223 (d) 6209
10. 10.What is the smallest number you must add to 97 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
11. 11.Which of these numbers is divisible by 4? (a) 3237 (b) 3231 (c) 3232 (d) 3234
12. 12.What is the smallest number you must add to 217 to make it divisible by 11? (a) 4 (b) 3 (c) 8 (d) 2
13. 13.What is the units digit of 3^14? (a) 3 (b) 9 (c) 7 (d) 1
14. 14.What is the units digit of 7^15? (a) 3 (b) 7 (c) 1 (d) 9
15. 15.What is the units digit of 2^38? (a) 8 (b) 2 (c) 4 (d) 6
16. 16.What is the units digit of 9^59? (a) 1 (b) 8 (c) 9 (d) 7
17. 17.What is the remainder when 212 is divided by 3? (a) 2 (b) 1 (c) 0
18. 18.Which of these numbers is divisible by 6? (a) 925 (b) 939 (c) 930 (d) 931
19. 19.What is the remainder when 217 is divided by 6? (a) 1 (b) 0 (c) 2 (d) 5
20. 20.What is the remainder when 483 is divided by 11? (a) 9 (b) 10 (c) 8 (d) 1
21. 21.Which of these numbers is divisible by 8? (a) 6880 (b) 6891 (c) 6887 (d) 6876
22. 22.What is the units digit of 2^44? (a) 6 (b) 8 (c) 2 (d) 4
23. 23.What is the smallest number you must add to 277 to make it divisible by 12? (a) 10 (b) 1 (c) 11 (d) 9
24. 24.Which of these numbers is divisible by 3? (a) 2525 (b) 2528 (c) 2530 (d) 2526
25. 25.What is the remainder when 232 is divided by 3? (a) 0 (b) 1 (c) 2
26. 26.Which of these numbers is divisible by 8? (a) 952 (b) 963 (c) 956 (d) 950
27. 27.What is the units digit of 7^36? (a) 3 (b) 1 (c) 7 (d) 9
28. 28.What is the smallest number you must add to 139 to make it divisible by 3? (a) 2 (b) 1
29. 29.What is the smallest number you must add to 267 to make it divisible by 11? (a) 3 (b) 8 (c) 7 (d) 9
30. 30.Which of these numbers is divisible by 3? (a) 932 (b) 935 (c) 928 (d) 930
31. 31.What is the smallest number you must add to 258 to make it divisible by 12? (a) 7 (b) 5 (c) 6 (d) 8
32. 32.What is the units digit of 7^10? (a) 9 (b) 3 (c) 7 (d) 1
33. 33.What is the units digit of 8^58? (a) 8 (b) 2 (c) 4 (d) 6
34. 34.Which of these numbers is divisible by 3? (a) 1618 (b) 1617 (c) 1615 (d) 1622
35. 35.What is the units digit of 8^34? (a) 4 (b) 2 (c) 6 (d) 8
36. 36.What is the units digit of 8^26? (a) 4 (b) 2 (c) 6 (d) 8
37. 37.What is the remainder when 263 is divided by 6? (a) 1 (b) 5 (c) 3 (d) 4
38. 38.What is the remainder when 175 is divided by 3? (a) 0 (b) 1 (c) 2
39. 39.Which of these numbers is divisible by 8? (a) 3435 (b) 3440 (c) 3450 (d) 3447
40. 40.Which of these numbers is divisible by 4? (a) 206 (b) 209 (c) 208 (d) 214

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad