Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 999 · 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 6? (a) 637 (b) 647 (c) 636 (d) 633
2. 2.Which of these numbers is divisible by 6? (a) 999 (b) 1005 (c) 1011 (d) 1002
3. 3.What is the units digit of 3^54? (a) 7 (b) 9 (c) 3 (d) 1
4. 4.Which of these numbers is divisible by 9? (a) 3961 (b) 3972 (c) 3954 (d) 3960
5. 5.What is the smallest number you must add to 121 to make it divisible by 3? (a) 2 (b) 1
6. 6.Which of these numbers is divisible by 3? (a) 1579 (b) 1581 (c) 1586 (d) 1583
7. 7.What is the remainder when 106 is divided by 3? (a) 0 (b) 1 (c) 2
8. 8.What is the units digit of 2^22? (a) 6 (b) 2 (c) 8 (d) 4
9. 9.What is the smallest number you must add to 49 to make it divisible by 4? (a) 2 (b) 1 (c) 3
10. 10.Which of these numbers is divisible by 3? (a) 1706 (b) 1700 (c) 1702 (d) 1701
11. 11.Which of these numbers is divisible by 6? (a) 3208 (b) 3201 (c) 3215 (d) 3204
12. 12.What is the smallest number you must add to 83 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
13. 13.Which of these numbers is divisible by 9? (a) 2911 (b) 2916 (c) 2930 (d) 2917
14. 14.What is the units digit of 7^46? (a) 7 (b) 9 (c) 1 (d) 3
15. 15.What is the units digit of 9^10? (a) 1 (b) 0 (c) 2 (d) 9
16. 16.What is the units digit of 8^38? (a) 6 (b) 2 (c) 8 (d) 4
17. 17.Which of these numbers is divisible by 9? (a) 681 (b) 671 (c) 675 (d) 692
18. 18.What is the units digit of 2^60? (a) 4 (b) 6 (c) 8 (d) 2
19. 19.What is the units digit of 9^23? (a) 7 (b) 8 (c) 1 (d) 9
20. 20.Which of these numbers is divisible by 4? (a) 3114 (b) 3111 (c) 3119 (d) 3112
21. 21.What is the smallest number you must add to 668 to make it divisible by 12? (a) 4 (b) 3 (c) 8 (d) 5
22. 22.Which of these numbers is divisible by 3? (a) 1343 (b) 1341 (c) 1346 (d) 1340
23. 23.What is the remainder when 241 is divided by 3? (a) 1 (b) 0 (c) 2
24. 24.Which of these numbers is divisible by 9? (a) 6937 (b) 6941 (c) 6951 (d) 6939
25. 25.What is the smallest number you must add to 322 to make it divisible by 9? (a) 3 (b) 2 (c) 7 (d) 1
26. 26.What is the remainder when 193 is divided by 7? (a) 6 (b) 3 (c) 4 (d) 5
27. 27.What is the remainder when 95 is divided by 9? (a) 7 (b) 6 (c) 4 (d) 5
28. 28.What is the smallest number you must add to 548 to make it divisible by 10? (a) 8 (b) 2 (c) 3 (d) 1
29. 29.Which of these numbers is divisible by 3? (a) 1642 (b) 1639 (c) 1641 (d) 1645
30. 30.Which of these numbers is divisible by 6? (a) 2096 (b) 2085 (c) 2091 (d) 2088
31. 31.What is the units digit of 3^33? (a) 3 (b) 7 (c) 1 (d) 9
32. 32.What is the remainder when 277 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
33. 33.Which of these numbers is divisible by 4? (a) 2799 (b) 2800 (c) 2805 (d) 2802
34. 34.What is the remainder when 61 is divided by 5? (a) 2 (b) 4 (c) 0 (d) 1
35. 35.What is the smallest number you must add to 842 to make it divisible by 11? (a) 5 (b) 6 (c) 4 (d) 7
36. 36.What is the units digit of 8^16? (a) 6 (b) 8 (c) 4 (d) 2
37. 37.What is the units digit of 2^45? (a) 4 (b) 8 (c) 2 (d) 6
38. 38.What is the units digit of 8^53? (a) 4 (b) 6 (c) 2 (d) 8
39. 39.What is the units digit of 2^32? (a) 2 (b) 6 (c) 8 (d) 4
40. 40.Which of these numbers is divisible by 8? (a) 5088 (b) 5086 (c) 5094 (d) 5098

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