Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 50 · 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.Which of these numbers is divisible by 11? (a) 6423 (b) 6424 (c) 6440 (d) 6425
2. 2.What is the units digit of 9^29? (a) 9 (b) 8 (c) 7 (d) 1
3. 3.Which of these numbers is divisible by 6? (a) 1850 (b) 1847 (c) 1841 (d) 1842
4. 4.What is the remainder when 297 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
5. 5.Which of these numbers is divisible by 3? (a) 365 (b) 370 (c) 368 (d) 366
6. 6.What is the smallest number you must add to 77 to make it divisible by 4? (a) 2 (b) 1 (c) 3
7. 7.What is the units digit of 3^49? (a) 3 (b) 9 (c) 7 (d) 1
8. 8.What is the units digit of 7^17? (a) 3 (b) 9 (c) 7 (d) 1
9. 9.What is the remainder when 139 is divided by 6? (a) 0 (b) 5 (c) 2 (d) 1
10. 10.What is the remainder when 356 is divided by 6? (a) 3 (b) 1 (c) 4 (d) 2
11. 11.What is the remainder when 469 is divided by 9? (a) 8 (b) 1 (c) 0 (d) 2
12. 12.What is the smallest number you must add to 172 to make it divisible by 10? (a) 2 (b) 9 (c) 8 (d) 7
13. 13.Which of these numbers is divisible by 6? (a) 4992 (b) 4996 (c) 4988 (d) 5003
14. 14.What is the remainder when 250 is divided by 8? (a) 3 (b) 6 (c) 2 (d) 1
15. 15.What is the smallest number you must add to 243 to make it divisible by 10? (a) 7 (b) 6 (c) 8 (d) 3
16. 16.What is the units digit of 3^60? (a) 7 (b) 9 (c) 1 (d) 3
17. 17.What is the units digit of 7^30? (a) 7 (b) 1 (c) 3 (d) 9
18. 18.Which of these numbers is divisible by 3? (a) 1760 (b) 1763 (c) 1765 (d) 1761
19. 19.What is the units digit of 8^14? (a) 4 (b) 2 (c) 6 (d) 8
20. 20.What is the units digit of 4^59? (a) 6 (b) 4 (c) 3 (d) 5
21. 21.What is the remainder when 368 is divided by 10? (a) 2 (b) 7 (c) 9 (d) 8
22. 22.What is the remainder when 625 is divided by 12? (a) 0 (b) 11 (c) 2 (d) 1
23. 23.What is the units digit of 4^48? (a) 7 (b) 6 (c) 4 (d) 5
24. 24.What is the smallest number you must add to 482 to make it divisible by 8? (a) 2 (b) 5 (c) 6 (d) 7
25. 25.What is the units digit of 2^10? (a) 2 (b) 8 (c) 6 (d) 4
26. 26.What is the remainder when 369 is divided by 10? (a) 9 (b) 7 (c) 1 (d) 8
27. 27.What is the smallest number you must add to 495 to make it divisible by 8? (a) 7 (b) 1 (c) 3 (d) 2
28. 28.Which of these numbers is divisible by 6? (a) 5243 (b) 5249 (c) 5244 (d) 5251
29. 29.What is the smallest number you must add to 169 to make it divisible by 8? (a) 6 (b) 1 (c) 7 (d) 5
30. 30.What is the smallest number you must add to 552 to make it divisible by 9? (a) 3 (b) 5 (c) 6 (d) 7
31. 31.What is the smallest number you must add to 107 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
32. 32.What is the units digit of 3^33? (a) 1 (b) 7 (c) 9 (d) 3
33. 33.What is the remainder when 419 is divided by 9? (a) 5 (b) 6 (c) 4 (d) 7
34. 34.What is the remainder when 500 is divided by 7? (a) 5 (b) 3 (c) 2 (d) 4
35. 35.What is the smallest number you must add to 152 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
36. 36.What is the smallest number you must add to 38 to make it divisible by 3? (a) 1 (b) 2
37. 37.What is the smallest number you must add to 111 to make it divisible by 7? (a) 2 (b) 3 (c) 6 (d) 1
38. 38.What is the remainder when 34 is divided by 3? (a) 2 (b) 0 (c) 1
39. 39.What is the smallest number you must add to 98 to make it divisible by 4? (a) 1 (b) 3 (c) 2
40. 40.What is the remainder when 171 is divided by 5? (a) 0 (b) 4 (c) 1 (d) 2

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