Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 342 · 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 4^55? (a) 5 (b) 6 (c) 3 (d) 4
2. 2.What is the units digit of 9^38? (a) 2 (b) 1 (c) 0 (d) 9
3. 3.What is the smallest number you must add to 189 to make it divisible by 4? (a) 2 (b) 3 (c) 1
4. 4.What is the units digit of 3^11? (a) 7 (b) 1 (c) 3 (d) 9
5. 5.Which of these numbers is divisible by 4? (a) 1654 (b) 1657 (c) 1651 (d) 1652
6. 6.What is the smallest number you must add to 69 to make it divisible by 4? (a) 3 (b) 2 (c) 1
7. 7.What is the remainder when 397 is divided by 6? (a) 2 (b) 5 (c) 1 (d) 0
8. 8.Which of these numbers is divisible by 11? (a) 2418 (b) 2420 (c) 2432 (d) 2424
9. 9.What is the units digit of 3^25? (a) 3 (b) 1 (c) 7 (d) 9
10. 10.Which of these numbers is divisible by 3? (a) 1437 (b) 1442 (c) 1436 (d) 1439
11. 11.Which of these numbers is divisible by 6? (a) 1011 (b) 1014 (c) 1023 (d) 1016
12. 12.Which of these numbers is divisible by 8? (a) 4753 (b) 4752 (c) 4767 (d) 4750
13. 13.What is the units digit of 8^32? (a) 4 (b) 8 (c) 2 (d) 6
14. 14.What is the smallest number you must add to 342 to make it divisible by 8? (a) 1 (b) 2 (c) 6 (d) 3
15. 15.Which of these numbers is divisible by 3? (a) 1892 (b) 1888 (c) 1887 (d) 1886
16. 16.Which of these numbers is divisible by 9? (a) 2648 (b) 2645 (c) 2634 (d) 2637
17. 17.What is the remainder when 196 is divided by 3? (a) 2 (b) 0 (c) 1
18. 18.What is the smallest number you must add to 323 to make it divisible by 8? (a) 5 (b) 6 (c) 3 (d) 4
19. 19.What is the units digit of 2^10? (a) 8 (b) 4 (c) 6 (d) 2
20. 20.What is the units digit of 7^54? (a) 1 (b) 9 (c) 7 (d) 3
21. 21.Which of these numbers is divisible by 8? (a) 2917 (b) 2903 (c) 2910 (d) 2904
22. 22.What is the smallest number you must add to 422 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 4
23. 23.Which of these numbers is divisible by 8? (a) 1604 (b) 1614 (c) 1599 (d) 1600
24. 24.Which of these numbers is divisible by 3? (a) 1793 (b) 1788 (c) 1786 (d) 1790
25. 25.What is the remainder when 98 is divided by 3? (a) 0 (b) 2 (c) 1
26. 26.Which of these numbers is divisible by 11? (a) 8131 (b) 8161 (c) 8148 (d) 8140
27. 27.Which of these numbers is divisible by 6? (a) 1431 (b) 1423 (c) 1437 (d) 1428
28. 28.What is the units digit of 7^36? (a) 7 (b) 9 (c) 3 (d) 1
29. 29.Which of these numbers is divisible by 9? (a) 1709 (b) 1699 (c) 1684 (d) 1692
30. 30.Which of these numbers is divisible by 6? (a) 1626 (b) 1624 (c) 1631 (d) 1635
31. 31.What is the remainder when 375 is divided by 9? (a) 3 (b) 5 (c) 6 (d) 7
32. 32.Which of these numbers is divisible by 4? (a) 1713 (b) 1718 (c) 1712 (d) 1709
33. 33.What is the remainder when 113 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
34. 34.What is the remainder when 111 is divided by 9? (a) 4 (b) 6 (c) 3 (d) 2
35. 35.What is the units digit of 4^45? (a) 5 (b) 3 (c) 4 (d) 6
36. 36.Which of these numbers is divisible by 9? (a) 1983 (b) 2000 (c) 1989 (d) 1992
37. 37.What is the smallest number you must add to 182 to make it divisible by 9? (a) 6 (b) 8 (c) 2 (d) 7
38. 38.Which of these numbers is divisible by 9? (a) 6176 (b) 6200 (c) 6185 (d) 6183
39. 39.Which of these numbers is divisible by 4? (a) 1981 (b) 1976 (c) 1977 (d) 1973
40. 40.Which of these numbers is divisible by 9? (a) 1091 (b) 1098 (c) 1115 (d) 1105

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