Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 599 · 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 9? (a) 2617 (b) 2626 (c) 2619 (d) 2635
2. 2.Which of these numbers is divisible by 4? (a) 2862 (b) 2854 (c) 2859 (d) 2856
3. 3.What is the units digit of 3^22? (a) 1 (b) 3 (c) 9 (d) 7
4. 4.Which of these numbers is divisible by 4? (a) 1959 (b) 1955 (c) 1962 (d) 1956
5. 5.Which of these numbers is divisible by 8? (a) 1682 (b) 1693 (c) 1698 (d) 1688
6. 6.What is the remainder when 118 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
7. 7.What is the units digit of 9^16? (a) 1 (b) 2 (c) 9 (d) 0
8. 8.Which of these numbers is divisible by 11? (a) 7264 (b) 7271 (c) 7286 (d) 7275
9. 9.What is the remainder when 679 is divided by 10? (a) 9 (b) 8 (c) 1 (d) 7
10. 10.Which of these numbers is divisible by 3? (a) 1192 (b) 1196 (c) 1190 (d) 1191
11. 11.What is the remainder when 585 is divided by 8? (a) 2 (b) 0 (c) 7 (d) 1
12. 12.What is the units digit of 4^38? (a) 5 (b) 7 (c) 4 (d) 6
13. 13.What is the remainder when 127 is divided by 11? (a) 8 (b) 5 (c) 6 (d) 7
14. 14.What is the units digit of 9^17? (a) 1 (b) 9 (c) 8 (d) 7
15. 15.What is the units digit of 8^22? (a) 2 (b) 8 (c) 6 (d) 4
16. 16.What is the units digit of 4^45? (a) 4 (b) 3 (c) 6 (d) 5
17. 17.Which of these numbers is divisible by 9? (a) 1132 (b) 1125 (c) 1121 (d) 1141
18. 18.What is the units digit of 3^55? (a) 3 (b) 1 (c) 7 (d) 9
19. 19.Which of these numbers is divisible by 11? (a) 5035 (b) 5056 (c) 5041 (d) 5038
20. 20.Which of these numbers is divisible by 6? (a) 851 (b) 854 (c) 860 (d) 852
21. 21.What is the remainder when 324 is divided by 10? (a) 3 (b) 4 (c) 5 (d) 6
22. 22.What is the remainder when 654 is divided by 10? (a) 5 (b) 4 (c) 3 (d) 6
23. 23.What is the units digit of 7^20? (a) 9 (b) 3 (c) 7 (d) 1
24. 24.What is the smallest number you must add to 861 to make it divisible by 11? (a) 3 (b) 7 (c) 9 (d) 8
25. 25.What is the remainder when 688 is divided by 10? (a) 7 (b) 9 (c) 2 (d) 8
26. 26.Which of these numbers is divisible by 3? (a) 1393 (b) 1397 (c) 1392 (d) 1390
27. 27.Which of these numbers is divisible by 11? (a) 9594 (b) 9609 (c) 9603 (d) 9623
28. 28.What is the units digit of 4^17? (a) 4 (b) 5 (c) 6 (d) 3
29. 29.What is the remainder when 505 is divided by 8? (a) 7 (b) 0 (c) 2 (d) 1
30. 30.What is the remainder when 485 is divided by 7? (a) 2 (b) 1 (c) 5 (d) 3
31. 31.What is the smallest number you must add to 182 to make it divisible by 3? (a) 2 (b) 1
32. 32.What is the units digit of 7^48? (a) 9 (b) 7 (c) 1 (d) 3
33. 33.What is the smallest number you must add to 131 to make it divisible by 4? (a) 1 (b) 3 (c) 2
34. 34.What is the smallest number you must add to 131 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
35. 35.What is the smallest number you must add to 622 to make it divisible by 10? (a) 2 (b) 9 (c) 7 (d) 8
36. 36.What is the smallest number you must add to 241 to make it divisible by 12? (a) 1 (b) 9 (c) 11 (d) 10
37. 37.Which of these numbers is divisible by 3? (a) 969 (b) 974 (c) 971 (d) 968
38. 38.What is the units digit of 8^40? (a) 2 (b) 6 (c) 8 (d) 4
39. 39.Which of these numbers is divisible by 6? (a) 943 (b) 931 (c) 936 (d) 940
40. 40.What is the smallest number you must add to 646 to make it divisible by 12? (a) 1 (b) 2 (c) 3 (d) 10

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