Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 341 · 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 7^52? (a) 7 (b) 9 (c) 3 (d) 1
2. 2.What is the smallest number you must add to 719 to make it divisible by 12? (a) 11 (b) 1 (c) 3 (d) 2
3. 3.What is the remainder when 140 is divided by 3? (a) 0 (b) 2 (c) 1
4. 4.What is the units digit of 3^26? (a) 7 (b) 9 (c) 3 (d) 1
5. 5.What is the units digit of 4^33? (a) 5 (b) 4 (c) 6 (d) 3
6. 6.What is the smallest number you must add to 402 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
7. 7.What is the smallest number you must add to 201 to make it divisible by 10? (a) 7 (b) 8 (c) 1 (d) 9
8. 8.Which of these numbers is divisible by 3? (a) 982 (b) 976 (c) 980 (d) 978
9. 9.What is the units digit of 7^19? (a) 7 (b) 1 (c) 3 (d) 9
10. 10.What is the units digit of 7^10? (a) 1 (b) 9 (c) 3 (d) 7
11. 11.Which of these numbers is divisible by 3? (a) 2479 (b) 2486 (c) 2483 (d) 2481
12. 12.What is the remainder when 445 is divided by 10? (a) 6 (b) 4 (c) 7 (d) 5
13. 13.Which of these numbers is divisible by 9? (a) 1743 (b) 1736 (c) 1749 (d) 1737
14. 14.Which of these numbers is divisible by 11? (a) 1898 (b) 1892 (c) 1885 (d) 1908
15. 15.What is the smallest number you must add to 278 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
16. 16.What is the smallest number you must add to 78 to make it divisible by 4? (a) 3 (b) 1 (c) 2
17. 17.What is the smallest number you must add to 409 to make it divisible by 9? (a) 4 (b) 7 (c) 6 (d) 5
18. 18.What is the remainder when 291 is divided by 5? (a) 0 (b) 2 (c) 4 (d) 1
19. 19.What is the remainder when 233 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
20. 20.What is the remainder when 808 is divided by 10? (a) 8 (b) 9 (c) 2 (d) 7
21. 21.Which of these numbers is divisible by 8? (a) 7004 (b) 7010 (c) 7000 (d) 6996
22. 22.What is the remainder when 544 is divided by 12? (a) 5 (b) 3 (c) 8 (d) 4
23. 23.What is the units digit of 9^15? (a) 1 (b) 9 (c) 7 (d) 8
24. 24.What is the smallest number you must add to 398 to make it divisible by 12? (a) 11 (b) 9 (c) 2 (d) 10
25. 25.What is the remainder when 58 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
26. 26.What is the smallest number you must add to 405 to make it divisible by 8? (a) 5 (b) 3 (c) 4 (d) 2
27. 27.Which of these numbers is divisible by 6? (a) 1454 (b) 1465 (c) 1458 (d) 1459
28. 28.What is the smallest number you must add to 307 to make it divisible by 4? (a) 2 (b) 1 (c) 3
29. 29.What is the remainder when 540 is divided by 7? (a) 6 (b) 2 (c) 0 (d) 1
30. 30.Which of these numbers is divisible by 6? (a) 5346 (b) 5351 (c) 5357 (d) 5343
31. 31.What is the units digit of 4^45? (a) 4 (b) 3 (c) 5 (d) 6
32. 32.Which of these numbers is divisible by 9? (a) 5793 (b) 5778 (c) 5776 (d) 5784
33. 33.What is the units digit of 3^42? (a) 3 (b) 1 (c) 9 (d) 7
34. 34.Which of these numbers is divisible by 4? (a) 875 (b) 876 (c) 879 (d) 881
35. 35.What is the remainder when 213 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
36. 36.What is the smallest number you must add to 470 to make it divisible by 9? (a) 6 (b) 7 (c) 8 (d) 2
37. 37.What is the smallest number you must add to 226 to make it divisible by 4? (a) 1 (b) 2 (c) 3
38. 38.What is the smallest number you must add to 226 to make it divisible by 9? (a) 8 (b) 1 (c) 6 (d) 7
39. 39.What is the units digit of 8^36? (a) 2 (b) 8 (c) 4 (d) 6
40. 40.What is the remainder when 447 is divided by 12? (a) 2 (b) 3 (c) 4 (d) 9

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