Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 735 · 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 246 to make it divisible by 12? (a) 7 (b) 5 (c) 8 (d) 6
2. 2.What is the units digit of 3^42? (a) 7 (b) 1 (c) 3 (d) 9
3. 3.What is the smallest number you must add to 171 to make it divisible by 4? (a) 2 (b) 3 (c) 1
4. 4.What is the units digit of 2^37? (a) 2 (b) 4 (c) 8 (d) 6
5. 5.What is the units digit of 4^49? (a) 3 (b) 4 (c) 6 (d) 5
6. 6.What is the units digit of 3^41? (a) 9 (b) 1 (c) 3 (d) 7
7. 7.What is the units digit of 9^22? (a) 9 (b) 0 (c) 2 (d) 1
8. 8.What is the remainder when 516 is divided by 7? (a) 6 (b) 2 (c) 5 (d) 4
9. 9.Which of these numbers is divisible by 11? (a) 2073 (b) 2088 (c) 2067 (d) 2068
10. 10.What is the remainder when 241 is divided by 8? (a) 2 (b) 1 (c) 0 (d) 7
11. 11.Which of these numbers is divisible by 9? (a) 6113 (b) 6094 (c) 6102 (d) 6109
12. 12.What is the remainder when 488 is divided by 9? (a) 2 (b) 3 (c) 1 (d) 7
13. 13.What is the remainder when 212 is divided by 11? (a) 8 (b) 2 (c) 4 (d) 3
14. 14.What is the units digit of 9^42? (a) 1 (b) 0 (c) 2 (d) 9
15. 15.What is the remainder when 213 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
16. 16.What is the remainder when 202 is divided by 8? (a) 2 (b) 3 (c) 6 (d) 1
17. 17.What is the remainder when 244 is divided by 12? (a) 3 (b) 5 (c) 4 (d) 8
18. 18.What is the remainder when 203 is divided by 3? (a) 0 (b) 2 (c) 1
19. 19.What is the units digit of 8^19? (a) 4 (b) 8 (c) 6 (d) 2
20. 20.Which of these numbers is divisible by 4? (a) 755 (b) 757 (c) 756 (d) 763
21. 21.What is the units digit of 7^24? (a) 7 (b) 3 (c) 9 (d) 1
22. 22.What is the smallest number you must add to 749 to make it divisible by 10? (a) 3 (b) 2 (c) 9 (d) 1
23. 23.What is the units digit of 2^16? (a) 2 (b) 4 (c) 8 (d) 6
24. 24.What is the remainder when 296 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 0
25. 25.Which of these numbers is divisible by 3? (a) 2212 (b) 2216 (c) 2218 (d) 2214
26. 26.What is the remainder when 546 is divided by 8? (a) 1 (b) 6 (c) 3 (d) 2
27. 27.Which of these numbers is divisible by 6? (a) 1491 (b) 1499 (c) 1494 (d) 1505
28. 28.What is the smallest number you must add to 289 to make it divisible by 4? (a) 1 (b) 3 (c) 2
29. 29.What is the smallest number you must add to 167 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
30. 30.Which of these numbers is divisible by 11? (a) 7463 (b) 7470 (c) 7485 (d) 7469
31. 31.What is the remainder when 202 is divided by 9? (a) 5 (b) 6 (c) 4 (d) 3
32. 32.What is the remainder when 215 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
33. 33.What is the units digit of 7^29? (a) 7 (b) 9 (c) 1 (d) 3
34. 34.What is the smallest number you must add to 173 to make it divisible by 7? (a) 1 (b) 3 (c) 5 (d) 2
35. 35.What is the units digit of 2^56? (a) 6 (b) 8 (c) 4 (d) 2
36. 36.Which of these numbers is divisible by 4? (a) 2819 (b) 2813 (c) 2811 (d) 2812
37. 37.What is the units digit of 4^31? (a) 3 (b) 4 (c) 6 (d) 5
38. 38.What is the smallest number you must add to 201 to make it divisible by 4? (a) 2 (b) 1 (c) 3
39. 39.What is the units digit of 8^32? (a) 8 (b) 4 (c) 2 (d) 6
40. 40.What is the units digit of 4^27? (a) 5 (b) 4 (c) 3 (d) 6

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