Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 714 · 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 smallest number you must add to 604 to make it divisible by 10? (a) 6 (b) 4 (c) 7 (d) 5
2. 2.Which of these numbers is divisible by 3? (a) 635 (b) 628 (c) 630 (d) 632
3. 3.What is the units digit of 7^28? (a) 1 (b) 9 (c) 7 (d) 3
4. 4.What is the units digit of 2^13? (a) 2 (b) 6 (c) 8 (d) 4
5. 5.What is the units digit of 7^16? (a) 9 (b) 7 (c) 3 (d) 1
6. 6.What is the units digit of 3^23? (a) 7 (b) 9 (c) 3 (d) 1
7. 7.What is the units digit of 9^49? (a) 8 (b) 9 (c) 1 (d) 7
8. 8.Which of these numbers is divisible by 4? (a) 3513 (b) 3521 (c) 3517 (d) 3516
9. 9.What is the remainder when 143 is divided by 10? (a) 2 (b) 3 (c) 4 (d) 7
10. 10.What is the smallest number you must add to 862 to make it divisible by 11? (a) 8 (b) 4 (c) 6 (d) 7
11. 11.Which of these numbers is divisible by 3? (a) 170 (b) 175 (c) 171 (d) 173
12. 12.What is the remainder when 465 is divided by 11? (a) 2 (b) 4 (c) 3 (d) 8
13. 13.Which of these numbers is divisible by 4? (a) 2990 (b) 2994 (c) 2998 (d) 2992
14. 14.Which of these numbers is divisible by 4? (a) 2771 (b) 2772 (c) 2773 (d) 2779
15. 15.What is the remainder when 505 is divided by 9? (a) 0 (b) 8 (c) 1 (d) 2
16. 16.Which of these numbers is divisible by 11? (a) 5718 (b) 5736 (c) 5720 (d) 5722
17. 17.What is the remainder when 46 is divided by 3? (a) 1 (b) 2 (c) 0
18. 18.What is the units digit of 3^14? (a) 3 (b) 9 (c) 1 (d) 7
19. 19.Which of these numbers is divisible by 3? (a) 955 (b) 961 (c) 959 (d) 957
20. 20.What is the units digit of 8^20? (a) 2 (b) 4 (c) 6 (d) 8
21. 21.What is the smallest number you must add to 80 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 5
22. 22.What is the smallest number you must add to 193 to make it divisible by 8? (a) 5 (b) 1 (c) 6 (d) 7
23. 23.Which of these numbers is divisible by 11? (a) 2097 (b) 2110 (c) 2101 (d) 2122
24. 24.What is the units digit of 4^13? (a) 5 (b) 3 (c) 4 (d) 6
25. 25.What is the remainder when 118 is divided by 3? (a) 1 (b) 2 (c) 0
26. 26.What is the remainder when 138 is divided by 10? (a) 9 (b) 8 (c) 2 (d) 7
27. 27.What is the units digit of 4^47? (a) 4 (b) 6 (c) 5 (d) 3
28. 28.Which of these numbers is divisible by 4? (a) 943 (b) 944 (c) 947 (d) 951
29. 29.What is the smallest number you must add to 110 to make it divisible by 3? (a) 1 (b) 2
30. 30.Which of these numbers is divisible by 9? (a) 7981 (b) 7985 (c) 7969 (d) 7974
31. 31.Which of these numbers is divisible by 11? (a) 2051 (b) 2041 (c) 2046 (d) 2059
32. 32.Which of these numbers is divisible by 4? (a) 1102 (b) 1096 (c) 1097 (d) 1095
33. 33.Which of these numbers is divisible by 8? (a) 4800 (b) 4796 (c) 4810 (d) 4801
34. 34.What is the units digit of 2^47? (a) 2 (b) 6 (c) 8 (d) 4
35. 35.What is the units digit of 3^11? (a) 1 (b) 9 (c) 3 (d) 7
36. 36.Which of these numbers is divisible by 11? (a) 1666 (b) 1677 (c) 1659 (d) 1661
37. 37.What is the units digit of 2^46? (a) 6 (b) 8 (c) 2 (d) 4
38. 38.Which of these numbers is divisible by 9? (a) 6906 (b) 6899 (c) 6914 (d) 6903
39. 39.What is the remainder when 141 is divided by 10? (a) 2 (b) 9 (c) 0 (d) 1
40. 40.What is the units digit of 3^21? (a) 1 (b) 3 (c) 9 (d) 7

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