Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 909 · 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 7^50? (a) 1 (b) 3 (c) 9 (d) 7
2. 2.What is the smallest number you must add to 45 to make it divisible by 4? (a) 2 (b) 1 (c) 3
3. 3.What is the units digit of 7^33? (a) 1 (b) 7 (c) 3 (d) 9
4. 4.What is the remainder when 834 is divided by 11? (a) 2 (b) 9 (c) 10 (d) 8
5. 5.What is the units digit of 8^44? (a) 2 (b) 6 (c) 8 (d) 4
6. 6.Which of these numbers is divisible by 6? (a) 2052 (b) 2047 (c) 2054 (d) 2063
7. 7.Which of these numbers is divisible by 8? (a) 4981 (b) 4984 (c) 4994 (d) 4990
8. 8.Which of these numbers is divisible by 3? (a) 2178 (b) 2179 (c) 2183 (d) 2177
9. 9.Which of these numbers is divisible by 8? (a) 6664 (b) 6671 (c) 6673 (d) 6658
10. 10.Which of these numbers is divisible by 3? (a) 895 (b) 892 (c) 890 (d) 891
11. 11.What is the smallest number you must add to 328 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
12. 12.Which of these numbers is divisible by 3? (a) 1831 (b) 1828 (c) 1835 (d) 1830
13. 13.What is the smallest number you must add to 343 to make it divisible by 11? (a) 2 (b) 8 (c) 9 (d) 10
14. 14.What is the units digit of 7^26? (a) 7 (b) 1 (c) 9 (d) 3
15. 15.Which of these numbers is divisible by 4? (a) 2557 (b) 2552 (c) 2553 (d) 2549
16. 16.Which of these numbers is divisible by 6? (a) 886 (b) 882 (c) 879 (d) 892
17. 17.What is the units digit of 2^39? (a) 8 (b) 2 (c) 4 (d) 6
18. 18.Which of these numbers is divisible by 3? (a) 2307 (b) 2311 (c) 2305 (d) 2308
19. 19.What is the smallest number you must add to 149 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
20. 20.Which of these numbers is divisible by 4? (a) 1469 (b) 1472 (c) 1474 (d) 1479
21. 21.Which of these numbers is divisible by 6? (a) 2021 (b) 2031 (c) 2022 (d) 2026
22. 22.Which of these numbers is divisible by 8? (a) 2701 (b) 2687 (c) 2695 (d) 2688
23. 23.What is the smallest number you must add to 519 to make it divisible by 12? (a) 3 (b) 10 (c) 8 (d) 9
24. 24.Which of these numbers is divisible by 9? (a) 1378 (b) 1361 (c) 1368 (d) 1371
25. 25.What is the units digit of 7^44? (a) 9 (b) 3 (c) 7 (d) 1
26. 26.What is the units digit of 9^34? (a) 9 (b) 0 (c) 2 (d) 1
27. 27.What is the smallest number you must add to 219 to make it divisible by 10? (a) 3 (b) 9 (c) 1 (d) 2
28. 28.What is the remainder when 52 is divided by 3? (a) 2 (b) 0 (c) 1
29. 29.Which of these numbers is divisible by 4? (a) 3372 (b) 3374 (c) 3370 (d) 3377
30. 30.What is the remainder when 128 is divided by 10? (a) 7 (b) 2 (c) 8 (d) 9
31. 31.What is the units digit of 9^26? (a) 2 (b) 1 (c) 0 (d) 9
32. 32.Which of these numbers is divisible by 8? (a) 3578 (b) 3587 (c) 3594 (d) 3584
33. 33.Which of these numbers is divisible by 3? (a) 1105 (b) 1100 (c) 1101 (d) 1103
34. 34.What is the remainder when 485 is divided by 9? (a) 8 (b) 7 (c) 1 (d) 6
35. 35.What is the smallest number you must add to 689 to make it divisible by 11? (a) 4 (b) 3 (c) 5 (d) 7
36. 36.What is the smallest number you must add to 51 to make it divisible by 4? (a) 3 (b) 2 (c) 1
37. 37.What is the smallest number you must add to 107 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 2
38. 38.Which of these numbers is divisible by 3? (a) 533 (b) 530 (c) 536 (d) 531
39. 39.Which of these numbers is divisible by 6? (a) 1402 (b) 1407 (c) 1404 (d) 1413
40. 40.What is the smallest number you must add to 100 to make it divisible by 6? (a) 4 (b) 2 (c) 3 (d) 1

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