Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 903 · 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 7^14? (a) 9 (b) 3 (c) 7 (d) 1
2. 2.What is the remainder when 59 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
3. 3.What is the remainder when 164 is divided by 11? (a) 8 (b) 1 (c) 10 (d) 9
4. 4.Which of these numbers is divisible by 4? (a) 644 (b) 650 (c) 641 (d) 645
5. 5.What is the smallest number you must add to 218 to make it divisible by 7? (a) 5 (b) 6 (c) 4 (d) 1
6. 6.What is the remainder when 92 is divided by 8? (a) 3 (b) 6 (c) 4 (d) 5
7. 7.What is the smallest number you must add to 222 to make it divisible by 4? (a) 2 (b) 3 (c) 1
8. 8.What is the remainder when 95 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
9. 9.What is the units digit of 7^42? (a) 1 (b) 7 (c) 9 (d) 3
10. 10.Which of these numbers is divisible by 6? (a) 3132 (b) 3134 (c) 3127 (d) 3142
11. 11.What is the units digit of 9^57? (a) 8 (b) 1 (c) 7 (d) 9
12. 12.Which of these numbers is divisible by 9? (a) 3117 (b) 3124 (c) 3123 (d) 3136
13. 13.What is the smallest number you must add to 149 to make it divisible by 4? (a) 3 (b) 2 (c) 1
14. 14.What is the smallest number you must add to 218 to make it divisible by 3? (a) 2 (b) 1
15. 15.What is the remainder when 34 is divided by 3? (a) 2 (b) 0 (c) 1
16. 16.Which of these numbers is divisible by 8? (a) 1809 (b) 1801 (c) 1808 (d) 1821
17. 17.What is the remainder when 192 is divided by 10? (a) 3 (b) 1 (c) 2 (d) 8
18. 18.Which of these numbers is divisible by 8? (a) 5329 (b) 5336 (c) 5338 (d) 5345
19. 19.What is the remainder when 40 is divided by 3? (a) 2 (b) 0 (c) 1
20. 20.What is the units digit of 2^14? (a) 2 (b) 4 (c) 6 (d) 8
21. 21.What is the smallest number you must add to 321 to make it divisible by 4? (a) 2 (b) 3 (c) 1
22. 22.What is the smallest number you must add to 664 to make it divisible by 12? (a) 7 (b) 9 (c) 8 (d) 4
23. 23.Which of these numbers is divisible by 11? (a) 2262 (b) 2234 (c) 2244 (d) 2245
24. 24.What is the units digit of 2^35? (a) 2 (b) 8 (c) 6 (d) 4
25. 25.What is the remainder when 327 is divided by 12? (a) 4 (b) 3 (c) 9 (d) 2
26. 26.What is the units digit of 4^53? (a) 3 (b) 6 (c) 5 (d) 4
27. 27.What is the units digit of 8^29? (a) 8 (b) 6 (c) 2 (d) 4
28. 28.What is the units digit of 9^10? (a) 0 (b) 1 (c) 9 (d) 2
29. 29.What is the remainder when 789 is divided by 10? (a) 1 (b) 9 (c) 8 (d) 7
30. 30.What is the units digit of 4^50? (a) 7 (b) 5 (c) 4 (d) 6
31. 31.What is the smallest number you must add to 285 to make it divisible by 8? (a) 3 (b) 4 (c) 5 (d) 2
32. 32.What is the units digit of 8^37? (a) 2 (b) 6 (c) 8 (d) 4
33. 33.What is the smallest number you must add to 283 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
34. 34.What is the remainder when 194 is divided by 6? (a) 1 (b) 4 (c) 3 (d) 2
35. 35.What is the remainder when 269 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
36. 36.What is the units digit of 2^29? (a) 6 (b) 4 (c) 2 (d) 8
37. 37.What is the smallest number you must add to 556 to make it divisible by 12? (a) 9 (b) 4 (c) 8 (d) 7
38. 38.What is the smallest number you must add to 749 to make it divisible by 10? (a) 3 (b) 1 (c) 9 (d) 2
39. 39.Which of these numbers is divisible by 4? (a) 1639 (b) 1640 (c) 1641 (d) 1646
40. 40.Which of these numbers is divisible by 9? (a) 6968 (b) 6975 (c) 6977 (d) 6989

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