Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 687 · 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 units digit of 9^11? (a) 1 (b) 7 (c) 9 (d) 8
2. 2.Which of these numbers is divisible by 4? (a) 2932 (b) 2934 (c) 2939 (d) 2930
3. 3.What is the units digit of 4^39? (a) 4 (b) 6 (c) 5 (d) 3
4. 4.What is the units digit of 8^31? (a) 6 (b) 4 (c) 2 (d) 8
5. 5.What is the units digit of 2^16? (a) 8 (b) 6 (c) 2 (d) 4
6. 6.Which of these numbers is divisible by 8? (a) 3779 (b) 3794 (c) 3791 (d) 3784
7. 7.What is the units digit of 7^18? (a) 9 (b) 3 (c) 1 (d) 7
8. 8.Which of these numbers is divisible by 9? (a) 6419 (b) 6438 (c) 6426 (d) 6434
9. 9.What is the remainder when 146 is divided by 10? (a) 6 (b) 4 (c) 7 (d) 5
10. 10.What is the remainder when 254 is divided by 6? (a) 1 (b) 2 (c) 4 (d) 3
11. 11.What is the remainder when 283 is divided by 10? (a) 4 (b) 2 (c) 3 (d) 7
12. 12.What is the units digit of 4^53? (a) 6 (b) 4 (c) 5 (d) 3
13. 13.Which of these numbers is divisible by 9? (a) 7958 (b) 7965 (c) 7980 (d) 7966
14. 14.What is the smallest number you must add to 73 to make it divisible by 3? (a) 1 (b) 2
15. 15.What is the smallest number you must add to 596 to make it divisible by 9? (a) 7 (b) 6 (c) 2 (d) 8
16. 16.Which of these numbers is divisible by 11? (a) 4616 (b) 4590 (c) 4605 (d) 4598
17. 17.What is the remainder when 276 is divided by 7? (a) 5 (b) 2 (c) 4 (d) 3
18. 18.Which of these numbers is divisible by 11? (a) 4322 (b) 4331 (c) 4312 (d) 4311
19. 19.What is the units digit of 2^19? (a) 6 (b) 8 (c) 2 (d) 4
20. 20.What is the remainder when 103 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
21. 21.Which of these numbers is divisible by 4? (a) 222 (b) 231 (c) 227 (d) 224
22. 22.What is the smallest number you must add to 533 to make it divisible by 11? (a) 7 (b) 5 (c) 8 (d) 6
23. 23.Which of these numbers is divisible by 3? (a) 1943 (b) 1944 (c) 1946 (d) 1948
24. 24.What is the units digit of 2^34? (a) 4 (b) 2 (c) 6 (d) 8
25. 25.What is the remainder when 146 is divided by 5? (a) 0 (b) 1 (c) 2 (d) 4
26. 26.What is the smallest number you must add to 650 to make it divisible by 12? (a) 11 (b) 9 (c) 2 (d) 10
27. 27.Which of these numbers is divisible by 11? (a) 1103 (b) 1124 (c) 1111 (d) 1112
28. 28.Which of these numbers is divisible by 9? (a) 1283 (b) 1277 (c) 1262 (d) 1269
29. 29.What is the smallest number you must add to 335 to make it divisible by 11? (a) 6 (b) 5 (c) 8 (d) 7
30. 30.What is the remainder when 287 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
31. 31.What is the units digit of 2^28? (a) 2 (b) 4 (c) 6 (d) 8
32. 32.What is the remainder when 793 is divided by 10? (a) 2 (b) 3 (c) 7 (d) 4
33. 33.What is the remainder when 391 is divided by 5? (a) 0 (b) 2 (c) 1 (d) 4
34. 34.What is the remainder when 571 is divided by 10? (a) 0 (b) 2 (c) 9 (d) 1
35. 35.What is the smallest number you must add to 229 to make it divisible by 3? (a) 1 (b) 2
36. 36.What is the units digit of 3^28? (a) 1 (b) 7 (c) 9 (d) 3
37. 37.What is the smallest number you must add to 347 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
38. 38.Which of these numbers is divisible by 4? (a) 806 (b) 799 (c) 802 (d) 800
39. 39.What is the smallest number you must add to 673 to make it divisible by 10? (a) 3 (b) 6 (c) 7 (d) 8
40. 40.What is the remainder when 185 is divided by 11? (a) 2 (b) 8 (c) 10 (d) 9

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