Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 200 · 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 9^50? (a) 2 (b) 1 (c) 9 (d) 0
2. 2.Which of these numbers is divisible by 9? (a) 6124 (b) 6135 (c) 6139 (d) 6129
3. 3.What is the units digit of 3^19? (a) 7 (b) 9 (c) 1 (d) 3
4. 4.Which of these numbers is divisible by 8? (a) 317 (b) 321 (c) 331 (d) 320
5. 5.Which of these numbers is divisible by 6? (a) 3312 (b) 3309 (c) 3323 (d) 3317
6. 6.What is the remainder when 627 is divided by 9? (a) 7 (b) 3 (c) 6 (d) 5
7. 7.Which of these numbers is divisible by 11? (a) 7026 (b) 7029 (c) 7045 (d) 7032
8. 8.Which of these numbers is divisible by 4? (a) 2551 (b) 2541 (c) 2544 (d) 2545
9. 9.What is the units digit of 8^16? (a) 4 (b) 8 (c) 2 (d) 6
10. 10.What is the smallest number you must add to 293 to make it divisible by 4? (a) 2 (b) 3 (c) 1
11. 11.What is the units digit of 4^60? (a) 6 (b) 5 (c) 7 (d) 4
12. 12.What is the remainder when 221 is divided by 3? (a) 0 (b) 1 (c) 2
13. 13.Which of these numbers is divisible by 6? (a) 3745 (b) 3751 (c) 3760 (d) 3750
14. 14.What is the remainder when 58 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
15. 15.Which of these numbers is divisible by 3? (a) 1868 (b) 1866 (c) 1870 (d) 1865
16. 16.What is the remainder when 437 is divided by 9? (a) 5 (b) 6 (c) 4 (d) 7
17. 17.What is the remainder when 124 is divided by 9? (a) 2 (b) 7 (c) 8 (d) 6
18. 18.Which of these numbers is divisible by 6? (a) 1656 (b) 1652 (c) 1660 (d) 1665
19. 19.Which of these numbers is divisible by 6? (a) 4734 (b) 4742 (c) 4735 (d) 4733
20. 20.What is the smallest number you must add to 232 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
21. 21.What is the smallest number you must add to 106 to make it divisible by 4? (a) 1 (b) 3 (c) 2
22. 22.What is the remainder when 334 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
23. 23.What is the remainder when 76 is divided by 3? (a) 1 (b) 2 (c) 0
24. 24.What is the remainder when 537 is divided by 10? (a) 6 (b) 3 (c) 7 (d) 8
25. 25.What is the units digit of 9^58? (a) 9 (b) 2 (c) 0 (d) 1
26. 26.Which of these numbers is divisible by 8? (a) 607 (b) 595 (c) 591 (d) 592
27. 27.Which of these numbers is divisible by 8? (a) 4150 (b) 4154 (c) 4166 (d) 4152
28. 28.What is the units digit of 7^39? (a) 9 (b) 1 (c) 3 (d) 7
29. 29.Which of these numbers is divisible by 8? (a) 3172 (b) 3157 (c) 3160 (d) 3162
30. 30.Which of these numbers is divisible by 8? (a) 1811 (b) 1816 (c) 1827 (d) 1823
31. 31.Which of these numbers is divisible by 3? (a) 508 (b) 506 (c) 504 (d) 502
32. 32.What is the units digit of 7^17? (a) 3 (b) 1 (c) 9 (d) 7
33. 33.What is the smallest number you must add to 86 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 4
34. 34.What is the remainder when 164 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
35. 35.What is the remainder when 93 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
36. 36.What is the units digit of 3^55? (a) 9 (b) 1 (c) 3 (d) 7
37. 37.What is the remainder when 357 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
38. 38.Which of these numbers is divisible by 6? (a) 931 (b) 924 (c) 922 (d) 925
39. 39.Which of these numbers is divisible by 11? (a) 2255 (b) 2251 (c) 2270 (d) 2262
40. 40.What is the units digit of 8^54? (a) 4 (b) 2 (c) 8 (d) 6

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