Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 382 · 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 3? (a) 2204 (b) 2200 (c) 2199 (d) 2198
2. 2.What is the units digit of 4^36? (a) 4 (b) 6 (c) 5 (d) 7
3. 3.What is the remainder when 310 is divided by 11? (a) 3 (b) 9 (c) 2 (d) 1
4. 4.What is the smallest number you must add to 122 to make it divisible by 9? (a) 3 (b) 4 (c) 6 (d) 5
5. 5.What is the units digit of 8^19? (a) 8 (b) 4 (c) 6 (d) 2
6. 6.What is the units digit of 7^12? (a) 3 (b) 9 (c) 7 (d) 1
7. 7.What is the smallest number you must add to 127 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
8. 8.What is the smallest number you must add to 99 to make it divisible by 8? (a) 3 (b) 5 (c) 6 (d) 4
9. 9.Which of these numbers is divisible by 8? (a) 5674 (b) 5670 (c) 5672 (d) 5684
10. 10.What is the remainder when 247 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
11. 11.What is the smallest number you must add to 700 to make it divisible by 11? (a) 7 (b) 4 (c) 5 (d) 3
12. 12.What is the units digit of 7^38? (a) 1 (b) 3 (c) 9 (d) 7
13. 13.What is the smallest number you must add to 129 to make it divisible by 10? (a) 1 (b) 2 (c) 3 (d) 9
14. 14.What is the units digit of 2^21? (a) 2 (b) 6 (c) 8 (d) 4
15. 15.What is the remainder when 169 is divided by 3? (a) 0 (b) 2 (c) 1
16. 16.Which of these numbers is divisible by 6? (a) 422 (b) 426 (c) 427 (d) 436
17. 17.What is the smallest number you must add to 256 to make it divisible by 11? (a) 3 (b) 9 (c) 8 (d) 7
18. 18.Which of these numbers is divisible by 8? (a) 1984 (b) 1977 (c) 1996 (d) 1985
19. 19.What is the units digit of 4^50? (a) 6 (b) 4 (c) 7 (d) 5
20. 20.What is the remainder when 88 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
21. 21.Which of these numbers is divisible by 3? (a) 1605 (b) 1610 (c) 1603 (d) 1607
22. 22.Which of these numbers is divisible by 4? (a) 3086 (b) 3091 (c) 3094 (d) 3088
23. 23.Which of these numbers is divisible by 4? (a) 1535 (b) 1531 (c) 1532 (d) 1538
24. 24.What is the units digit of 9^59? (a) 9 (b) 1 (c) 8 (d) 7
25. 25.What is the smallest number you must add to 58 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
26. 26.What is the units digit of 4^55? (a) 5 (b) 4 (c) 6 (d) 3
27. 27.Which of these numbers is divisible by 11? (a) 1750 (b) 1732 (c) 1738 (d) 1740
28. 28.What is the units digit of 7^29? (a) 9 (b) 1 (c) 7 (d) 3
29. 29.What is the units digit of 9^55? (a) 1 (b) 8 (c) 9 (d) 7
30. 30.What is the remainder when 265 is divided by 12? (a) 11 (b) 1 (c) 0 (d) 2
31. 31.What is the smallest number you must add to 484 to make it divisible by 6? (a) 3 (b) 2 (c) 1 (d) 4
32. 32.What is the smallest number you must add to 431 to make it divisible by 7? (a) 4 (b) 5 (c) 2 (d) 3
33. 33.What is the smallest number you must add to 281 to make it divisible by 4? (a) 3 (b) 2 (c) 1
34. 34.What is the remainder when 138 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
35. 35.What is the remainder when 241 is divided by 8? (a) 2 (b) 1 (c) 0 (d) 7
36. 36.What is the units digit of 7^26? (a) 9 (b) 7 (c) 1 (d) 3
37. 37.Which of these numbers is divisible by 6? (a) 2969 (b) 2972 (c) 2964 (d) 2961
38. 38.What is the smallest number you must add to 493 to make it divisible by 10? (a) 8 (b) 6 (c) 3 (d) 7
39. 39.What is the units digit of 4^35? (a) 6 (b) 5 (c) 4 (d) 3
40. 40.What is the smallest number you must add to 107 to make it divisible by 3? (a) 2 (b) 1

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