Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 760 · 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 4^35? (a) 5 (b) 4 (c) 3 (d) 6
2. 2.Which of these numbers is divisible by 4? (a) 2898 (b) 2896 (c) 2894 (d) 2901
3. 3.What is the smallest number you must add to 312 to make it divisible by 7? (a) 3 (b) 2 (c) 4 (d) 5
4. 4.What is the smallest number you must add to 138 to make it divisible by 7? (a) 3 (b) 5 (c) 1 (d) 2
5. 5.Which of these numbers is divisible by 4? (a) 177 (b) 176 (c) 174 (d) 183
6. 6.What is the remainder when 110 is divided by 8? (a) 2 (b) 6 (c) 5 (d) 7
7. 7.What is the smallest number you must add to 56 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
8. 8.What is the remainder when 614 is divided by 12? (a) 2 (b) 10 (c) 1 (d) 3
9. 9.Which of these numbers is divisible by 8? (a) 3390 (b) 3381 (c) 3397 (d) 3384
10. 10.What is the remainder when 323 is divided by 8? (a) 4 (b) 3 (c) 2 (d) 5
11. 11.Which of these numbers is divisible by 9? (a) 2188 (b) 2184 (c) 2178 (d) 2176
12. 12.What is the remainder when 241 is divided by 5? (a) 0 (b) 2 (c) 1 (d) 4
13. 13.What is the smallest number you must add to 527 to make it divisible by 7? (a) 4 (b) 6 (c) 2 (d) 5
14. 14.What is the remainder when 169 is divided by 8? (a) 2 (b) 1 (c) 0 (d) 7
15. 15.What is the smallest number you must add to 283 to make it divisible by 8? (a) 4 (b) 3 (c) 5 (d) 6
16. 16.Which of these numbers is divisible by 4? (a) 3553 (b) 3558 (c) 3556 (d) 3562
17. 17.What is the remainder when 434 is divided by 8? (a) 1 (b) 3 (c) 2 (d) 6
18. 18.What is the smallest number you must add to 100 to make it divisible by 7? (a) 6 (b) 5 (c) 2 (d) 4
19. 19.Which of these numbers is divisible by 9? (a) 5301 (b) 5313 (c) 5305 (d) 5297
20. 20.What is the units digit of 2^58? (a) 6 (b) 4 (c) 8 (d) 2
21. 21.What is the remainder when 169 is divided by 12? (a) 2 (b) 11 (c) 0 (d) 1
22. 22.What is the remainder when 274 is divided by 9? (a) 4 (b) 5 (c) 6 (d) 3
23. 23.What is the remainder when 203 is divided by 11? (a) 4 (b) 6 (c) 7 (d) 5
24. 24.What is the smallest number you must add to 548 to make it divisible by 10? (a) 8 (b) 2 (c) 3 (d) 1
25. 25.What is the smallest number you must add to 706 to make it divisible by 11? (a) 9 (b) 10 (c) 8 (d) 2
26. 26.What is the units digit of 8^54? (a) 2 (b) 6 (c) 4 (d) 8
27. 27.Which of these numbers is divisible by 9? (a) 4971 (b) 4962 (c) 4968 (d) 4985
28. 28.What is the units digit of 3^41? (a) 7 (b) 1 (c) 3 (d) 9
29. 29.Which of these numbers is divisible by 3? (a) 1026 (b) 1024 (c) 1028 (d) 1031
30. 30.What is the remainder when 187 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
31. 31.What is the smallest number you must add to 359 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
32. 32.Which of these numbers is divisible by 3? (a) 622 (b) 624 (c) 628 (d) 626
33. 33.What is the smallest number you must add to 199 to make it divisible by 4? (a) 3 (b) 2 (c) 1
34. 34.What is the smallest number you must add to 579 to make it divisible by 8? (a) 6 (b) 5 (c) 4 (d) 3
35. 35.What is the remainder when 62 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
36. 36.Which of these numbers is divisible by 4? (a) 809 (b) 806 (c) 808 (d) 814
37. 37.What is the units digit of 8^48? (a) 8 (b) 4 (c) 6 (d) 2
38. 38.Which of these numbers is divisible by 8? (a) 4592 (b) 4598 (c) 4588 (d) 4602
39. 39.Which of these numbers is divisible by 11? (a) 1766 (b) 1771 (c) 1784 (d) 1775
40. 40.What is the units digit of 8^17? (a) 4 (b) 8 (c) 2 (d) 6

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