Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 139 · 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 7^36? (a) 1 (b) 7 (c) 3 (d) 9
2. 2.Which of these numbers is divisible by 9? (a) 5779 (b) 5789 (c) 5787 (d) 5800
3. 3.What is the remainder when 61 is divided by 5? (a) 4 (b) 2 (c) 0 (d) 1
4. 4.What is the remainder when 141 is divided by 5? (a) 2 (b) 0 (c) 4 (d) 1
5. 5.What is the smallest number you must add to 532 to make it divisible by 9? (a) 1 (b) 7 (c) 6 (d) 8
6. 6.What is the smallest number you must add to 121 to make it divisible by 4? (a) 1 (b) 3 (c) 2
7. 7.What is the remainder when 574 is divided by 8? (a) 6 (b) 2 (c) 7 (d) 5
8. 8.Which of these numbers is divisible by 4? (a) 2197 (b) 2194 (c) 2190 (d) 2192
9. 9.Which of these numbers is divisible by 3? (a) 1220 (b) 1213 (c) 1217 (d) 1215
10. 10.What is the units digit of 7^51? (a) 1 (b) 9 (c) 7 (d) 3
11. 11.What is the remainder when 462 is divided by 8? (a) 7 (b) 2 (c) 6 (d) 5
12. 12.What is the units digit of 7^26? (a) 1 (b) 7 (c) 9 (d) 3
13. 13.What is the units digit of 8^32? (a) 2 (b) 4 (c) 8 (d) 6
14. 14.What is the smallest number you must add to 823 to make it divisible by 11? (a) 3 (b) 1 (c) 2 (d) 9
15. 15.Which of these numbers is divisible by 9? (a) 7483 (b) 7488 (c) 7501 (d) 7492
16. 16.What is the smallest number you must add to 253 to make it divisible by 12? (a) 10 (b) 11 (c) 9 (d) 1
17. 17.What is the smallest number you must add to 84 to make it divisible by 8? (a) 4 (b) 3 (c) 5 (d) 6
18. 18.What is the smallest number you must add to 241 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
19. 19.What is the smallest number you must add to 58 to make it divisible by 3? (a) 1 (b) 2
20. 20.What is the smallest number you must add to 391 to make it divisible by 12? (a) 7 (b) 6 (c) 5 (d) 4
21. 21.What is the units digit of 8^45? (a) 2 (b) 6 (c) 8 (d) 4
22. 22.What is the remainder when 84 is divided by 8? (a) 5 (b) 3 (c) 6 (d) 4
23. 23.What is the remainder when 157 is divided by 9? (a) 3 (b) 5 (c) 6 (d) 4
24. 24.What is the units digit of 2^19? (a) 2 (b) 8 (c) 4 (d) 6
25. 25.What is the units digit of 2^40? (a) 6 (b) 4 (c) 8 (d) 2
26. 26.Which of these numbers is divisible by 4? (a) 2930 (b) 2928 (c) 2935 (d) 2926
27. 27.What is the remainder when 131 is divided by 3? (a) 0 (b) 2 (c) 1
28. 28.Which of these numbers is divisible by 6? (a) 2888 (b) 2903 (c) 2892 (d) 2897
29. 29.What is the smallest number you must add to 92 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
30. 30.Which of these numbers is divisible by 6? (a) 4765 (b) 4770 (c) 4777 (d) 4774
31. 31.Which of these numbers is divisible by 3? (a) 859 (b) 861 (c) 865 (d) 863
32. 32.Which of these numbers is divisible by 6? (a) 4476 (b) 4475 (c) 4481 (d) 4485
33. 33.What is the units digit of 7^12? (a) 7 (b) 3 (c) 9 (d) 1
34. 34.Which of these numbers is divisible by 8? (a) 4174 (b) 4156 (c) 4160 (d) 4165
35. 35.What is the remainder when 286 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
36. 36.What is the units digit of 9^41? (a) 1 (b) 8 (c) 7 (d) 9
37. 37.What is the smallest number you must add to 347 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
38. 38.What is the smallest number you must add to 590 to make it divisible by 9? (a) 3 (b) 5 (c) 4 (d) 6
39. 39.What is the smallest number you must add to 266 to make it divisible by 4? (a) 3 (b) 2 (c) 1
40. 40.What is the units digit of 2^10? (a) 8 (b) 4 (c) 6 (d) 2

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