Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 102 · 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 257 is divided by 6? (a) 5 (b) 4 (c) 3 (d) 1
2. 2.What is the units digit of 8^45? (a) 8 (b) 6 (c) 4 (d) 2
3. 3.What is the smallest number you must add to 771 to make it divisible by 11? (a) 9 (b) 10 (c) 8 (d) 1
4. 4.What is the units digit of 3^13? (a) 3 (b) 9 (c) 1 (d) 7
5. 5.Which of these numbers is divisible by 8? (a) 5947 (b) 5953 (c) 5952 (d) 5962
6. 6.What is the remainder when 715 is divided by 12? (a) 5 (b) 7 (c) 8 (d) 6
7. 7.What is the units digit of 2^42? (a) 8 (b) 4 (c) 2 (d) 6
8. 8.Which of these numbers is divisible by 8? (a) 4967 (b) 4970 (c) 4953 (d) 4960
9. 9.What is the remainder when 570 is divided by 8? (a) 3 (b) 1 (c) 6 (d) 2
10. 10.Which of these numbers is divisible by 8? (a) 596 (b) 581 (c) 584 (d) 590
11. 11.Which of these numbers is divisible by 6? (a) 1627 (b) 1636 (c) 1622 (d) 1626
12. 12.What is the smallest number you must add to 463 to make it divisible by 6? (a) 4 (b) 3 (c) 5 (d) 1
13. 13.Which of these numbers is divisible by 3? (a) 2609 (b) 2612 (c) 2607 (d) 2606
14. 14.What is the smallest number you must add to 242 to make it divisible by 4? (a) 1 (b) 3 (c) 2
15. 15.Which of these numbers is divisible by 9? (a) 389 (b) 372 (c) 378 (d) 381
16. 16.What is the remainder when 659 is divided by 10? (a) 9 (b) 7 (c) 1 (d) 8
17. 17.What is the units digit of 9^17? (a) 7 (b) 1 (c) 8 (d) 9
18. 18.Which of these numbers is divisible by 11? (a) 3850 (b) 3853 (c) 3870 (d) 3842
19. 19.What is the smallest number you must add to 388 to make it divisible by 9? (a) 6 (b) 7 (c) 8 (d) 1
20. 20.What is the remainder when 373 is divided by 12? (a) 2 (b) 1 (c) 11 (d) 0
21. 21.What is the remainder when 127 is divided by 3? (a) 0 (b) 1 (c) 2
22. 22.What is the remainder when 234 is divided by 8? (a) 2 (b) 1 (c) 3 (d) 6
23. 23.Which of these numbers is divisible by 6? (a) 4403 (b) 4415 (c) 4406 (d) 4404
24. 24.Which of these numbers is divisible by 3? (a) 2680 (b) 2676 (c) 2675 (d) 2678
25. 25.Which of these numbers is divisible by 6? (a) 5040 (b) 5043 (c) 5047 (d) 5039
26. 26.What is the units digit of 4^47? (a) 4 (b) 5 (c) 3 (d) 6
27. 27.What is the remainder when 74 is divided by 6? (a) 1 (b) 3 (c) 2 (d) 4
28. 28.What is the remainder when 141 is divided by 5? (a) 0 (b) 4 (c) 2 (d) 1
29. 29.Which of these numbers is divisible by 6? (a) 575 (b) 584 (c) 577 (d) 576
30. 30.Which of these numbers is divisible by 3? (a) 1949 (b) 1947 (c) 1946 (d) 1951
31. 31.What is the smallest number you must add to 349 to make it divisible by 7? (a) 2 (b) 6 (c) 3 (d) 1
32. 32.Which of these numbers is divisible by 3? (a) 254 (b) 247 (c) 251 (d) 249
33. 33.What is the smallest number you must add to 370 to make it divisible by 8? (a) 5 (b) 2 (c) 6 (d) 7
34. 34.What is the remainder when 54 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
35. 35.Which of these numbers is divisible by 9? (a) 2657 (b) 2650 (c) 2639 (d) 2646
36. 36.Which of these numbers is divisible by 6? (a) 2065 (b) 2070 (c) 2079 (d) 2074
37. 37.What is the units digit of 9^28? (a) 1 (b) 9 (c) 2 (d) 0
38. 38.What is the units digit of 4^44? (a) 5 (b) 6 (c) 4 (d) 7
39. 39.What is the smallest number you must add to 153 to make it divisible by 6? (a) 3 (b) 5 (c) 2 (d) 4
40. 40.What is the units digit of 3^15? (a) 7 (b) 1 (c) 3 (d) 9

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