Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 891 · 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 remainder when 706 is divided by 11? (a) 9 (b) 3 (c) 2 (d) 1
2. 2.Which of these numbers is divisible by 6? (a) 901 (b) 893 (c) 894 (d) 897
3. 3.Which of these numbers is divisible by 11? (a) 1251 (b) 1230 (c) 1237 (d) 1232
4. 4.What is the smallest number you must add to 668 to make it divisible by 9? (a) 2 (b) 7 (c) 6 (d) 8
5. 5.Which of these numbers is divisible by 9? (a) 7488 (b) 7486 (c) 7492 (d) 7499
6. 6.What is the smallest number you must add to 111 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 4
7. 7.What is the remainder when 331 is divided by 10? (a) 0 (b) 2 (c) 1 (d) 9
8. 8.What is the remainder when 149 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
9. 9.Which of these numbers is divisible by 3? (a) 2519 (b) 2516 (c) 2514 (d) 2513
10. 10.What is the units digit of 4^31? (a) 3 (b) 4 (c) 6 (d) 5
11. 11.What is the units digit of 8^39? (a) 4 (b) 6 (c) 8 (d) 2
12. 12.What is the units digit of 4^42? (a) 7 (b) 4 (c) 6 (d) 5
13. 13.What is the remainder when 319 is divided by 10? (a) 9 (b) 8 (c) 7 (d) 1
14. 14.Which of these numbers is divisible by 11? (a) 5775 (b) 5778 (c) 5769 (d) 5791
15. 15.Which of these numbers is divisible by 3? (a) 1928 (b) 1930 (c) 1929 (d) 1934
16. 16.Which of these numbers is divisible by 4? (a) 365 (b) 359 (c) 360 (d) 363
17. 17.What is the remainder when 467 is divided by 9? (a) 8 (b) 1 (c) 6 (d) 7
18. 18.Which of these numbers is divisible by 3? (a) 718 (b) 715 (c) 713 (d) 714
19. 19.What is the remainder when 452 is divided by 7? (a) 6 (b) 4 (c) 3 (d) 5
20. 20.What is the units digit of 7^23? (a) 9 (b) 7 (c) 3 (d) 1
21. 21.Which of these numbers is divisible by 9? (a) 6697 (b) 6695 (c) 6696 (d) 6712
22. 22.What is the smallest number you must add to 567 to make it divisible by 8? (a) 2 (b) 3 (c) 1 (d) 7
23. 23.What is the smallest number you must add to 591 to make it divisible by 9? (a) 6 (b) 2 (c) 4 (d) 3
24. 24.What is the units digit of 7^24? (a) 1 (b) 9 (c) 3 (d) 7
25. 25.What is the smallest number you must add to 265 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 5
26. 26.Which of these numbers is divisible by 4? (a) 1915 (b) 1912 (c) 1918 (d) 1909
27. 27.Which of these numbers is divisible by 4? (a) 1956 (b) 1963 (c) 1958 (d) 1955
28. 28.What is the units digit of 9^39? (a) 8 (b) 1 (c) 7 (d) 9
29. 29.What is the smallest number you must add to 214 to make it divisible by 3? (a) 1 (b) 2
30. 30.Which of these numbers is divisible by 3? (a) 2428 (b) 2425 (c) 2427 (d) 2432
31. 31.What is the smallest number you must add to 422 to make it divisible by 12? (a) 11 (b) 10 (c) 2 (d) 9
32. 32.What is the remainder when 371 is divided by 8? (a) 4 (b) 5 (c) 3 (d) 2
33. 33.Which of these numbers is divisible by 11? (a) 7392 (b) 7401 (c) 7387 (d) 7407
34. 34.What is the smallest number you must add to 128 to make it divisible by 11? (a) 7 (b) 4 (c) 3 (d) 5
35. 35.Which of these numbers is divisible by 4? (a) 2370 (b) 2364 (c) 2365 (d) 2363
36. 36.What is the remainder when 267 is divided by 10? (a) 3 (b) 8 (c) 6 (d) 7
37. 37.What is the remainder when 262 is divided by 7? (a) 4 (b) 3 (c) 2 (d) 5
38. 38.Which of these numbers is divisible by 3? (a) 1800 (b) 1802 (c) 1805 (d) 1799
39. 39.What is the units digit of 7^45? (a) 1 (b) 9 (c) 3 (d) 7
40. 40.What is the smallest number you must add to 128 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1

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