Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 652 · 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 320 to make it divisible by 11? (a) 1 (b) 8 (c) 10 (d) 9
2. 2.What is the remainder when 111 is divided by 5? (a) 2 (b) 0 (c) 4 (d) 1
3. 3.What is the units digit of 3^10? (a) 3 (b) 7 (c) 9 (d) 1
4. 4.What is the remainder when 699 is divided by 9? (a) 6 (b) 5 (c) 3 (d) 7
5. 5.Which of these numbers is divisible by 9? (a) 6787 (b) 6796 (c) 6781 (d) 6786
6. 6.What is the units digit of 4^56? (a) 6 (b) 7 (c) 5 (d) 4
7. 7.What is the remainder when 171 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
8. 8.Which of these numbers is divisible by 9? (a) 2009 (b) 2017 (c) 2006 (d) 2007
9. 9.What is the remainder when 277 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
10. 10.What is the smallest number you must add to 246 to make it divisible by 7? (a) 1 (b) 5 (c) 4 (d) 6
11. 11.What is the units digit of 8^11? (a) 8 (b) 2 (c) 6 (d) 4
12. 12.What is the smallest number you must add to 131 to make it divisible by 4? (a) 3 (b) 2 (c) 1
13. 13.Which of these numbers is divisible by 11? (a) 2441 (b) 2442 (c) 2449 (d) 2456
14. 14.What is the smallest number you must add to 325 to make it divisible by 12? (a) 11 (b) 1 (c) 9 (d) 10
15. 15.What is the units digit of 8^48? (a) 4 (b) 2 (c) 8 (d) 6
16. 16.What is the units digit of 4^54? (a) 7 (b) 5 (c) 6 (d) 4
17. 17.Which of these numbers is divisible by 4? (a) 3375 (b) 3369 (c) 3366 (d) 3368
18. 18.What is the units digit of 2^18? (a) 4 (b) 2 (c) 8 (d) 6
19. 19.What is the smallest number you must add to 833 to make it divisible by 12? (a) 6 (b) 5 (c) 8 (d) 7
20. 20.What is the remainder when 793 is divided by 11? (a) 0 (b) 10 (c) 1 (d) 2
21. 21.Which of these numbers is divisible by 3? (a) 1103 (b) 1106 (c) 1099 (d) 1101
22. 22.Which of these numbers is divisible by 8? (a) 6755 (b) 6760 (c) 6775 (d) 6763
23. 23.What is the remainder when 375 is divided by 9? (a) 6 (b) 3 (c) 5 (d) 7
24. 24.What is the units digit of 7^37? (a) 9 (b) 3 (c) 7 (d) 1
25. 25.What is the remainder when 438 is divided by 11? (a) 2 (b) 10 (c) 8 (d) 9
26. 26.Which of these numbers is divisible by 4? (a) 2558 (b) 2552 (c) 2549 (d) 2553
27. 27.What is the remainder when 44 is divided by 3? (a) 0 (b) 1 (c) 2
28. 28.Which of these numbers is divisible by 3? (a) 173 (b) 171 (c) 175 (d) 170
29. 29.What is the units digit of 2^39? (a) 6 (b) 4 (c) 2 (d) 8
30. 30.What is the smallest number you must add to 279 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
31. 31.What is the remainder when 223 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
32. 32.What is the remainder when 952 is divided by 12? (a) 8 (b) 5 (c) 4 (d) 3
33. 33.What is the units digit of 3^59? (a) 7 (b) 3 (c) 9 (d) 1
34. 34.What is the smallest number you must add to 471 to make it divisible by 6? (a) 2 (b) 5 (c) 4 (d) 3
35. 35.Which of these numbers is divisible by 11? (a) 583 (b) 598 (c) 574 (d) 587
36. 36.Which of these numbers is divisible by 3? (a) 2431 (b) 2429 (c) 2430 (d) 2434
37. 37.What is the remainder when 336 is divided by 9? (a) 6 (b) 4 (c) 2 (d) 3
38. 38.What is the smallest number you must add to 125 to make it divisible by 9? (a) 8 (b) 3 (c) 2 (d) 1
39. 39.What is the units digit of 3^16? (a) 1 (b) 9 (c) 7 (d) 3
40. 40.What is the smallest number you must add to 830 to make it divisible by 12? (a) 9 (b) 2 (c) 10 (d) 11

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