Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 365 · 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 9? (a) 5058 (b) 5064 (c) 5068 (d) 5053
2. 2.What is the units digit of 8^17? (a) 2 (b) 8 (c) 4 (d) 6
3. 3.What is the remainder when 246 is divided by 11? (a) 5 (b) 7 (c) 4 (d) 3
4. 4.Which of these numbers is divisible by 3? (a) 1372 (b) 1369 (c) 1367 (d) 1368
5. 5.Which of these numbers is divisible by 4? (a) 1643 (b) 1636 (c) 1638 (d) 1634
6. 6.What is the remainder when 148 is divided by 11? (a) 6 (b) 7 (c) 5 (d) 4
7. 7.What is the smallest number you must add to 274 to make it divisible by 10? (a) 7 (b) 6 (c) 5 (d) 4
8. 8.What is the units digit of 9^60? (a) 2 (b) 9 (c) 1 (d) 0
9. 9.What is the units digit of 3^20? (a) 7 (b) 1 (c) 9 (d) 3
10. 10.Which of these numbers is divisible by 3? (a) 1937 (b) 1934 (c) 1939 (d) 1935
11. 11.What is the remainder when 254 is divided by 11? (a) 1 (b) 0 (c) 10 (d) 2
12. 12.What is the units digit of 3^59? (a) 3 (b) 1 (c) 7 (d) 9
13. 13.Which of these numbers is divisible by 9? (a) 7421 (b) 7431 (c) 7410 (d) 7416
14. 14.What is the smallest number you must add to 113 to make it divisible by 4? (a) 1 (b) 2 (c) 3
15. 15.What is the remainder when 70 is divided by 3? (a) 0 (b) 2 (c) 1
16. 16.What is the smallest number you must add to 700 to make it divisible by 11? (a) 7 (b) 5 (c) 3 (d) 4
17. 17.What is the units digit of 4^23? (a) 4 (b) 5 (c) 6 (d) 3
18. 18.What is the units digit of 2^21? (a) 4 (b) 6 (c) 2 (d) 8
19. 19.What is the remainder when 237 is divided by 10? (a) 3 (b) 7 (c) 8 (d) 6
20. 20.Which of these numbers is divisible by 6? (a) 2228 (b) 2224 (c) 2234 (d) 2226
21. 21.What is the smallest number you must add to 391 to make it divisible by 7? (a) 1 (b) 6 (c) 3 (d) 2
22. 22.What is the units digit of 7^47? (a) 1 (b) 9 (c) 3 (d) 7
23. 23.What is the units digit of 8^54? (a) 4 (b) 6 (c) 2 (d) 8
24. 24.What is the smallest number you must add to 257 to make it divisible by 6? (a) 5 (b) 1 (c) 2 (d) 3
25. 25.Which of these numbers is divisible by 4? (a) 2530 (b) 2526 (c) 2528 (d) 2535
26. 26.Which of these numbers is divisible by 3? (a) 1850 (b) 1848 (c) 1846 (d) 1853
27. 27.Which of these numbers is divisible by 8? (a) 1354 (b) 1361 (c) 1348 (d) 1352
28. 28.What is the remainder when 415 is divided by 9? (a) 8 (b) 2 (c) 1 (d) 0
29. 29.What is the units digit of 3^47? (a) 7 (b) 3 (c) 9 (d) 1
30. 30.What is the units digit of 9^40? (a) 0 (b) 9 (c) 2 (d) 1
31. 31.What is the units digit of 2^20? (a) 8 (b) 2 (c) 4 (d) 6
32. 32.Which of these numbers is divisible by 6? (a) 2968 (b) 2962 (c) 2974 (d) 2964
33. 33.What is the units digit of 3^54? (a) 3 (b) 1 (c) 7 (d) 9
34. 34.Which of these numbers is divisible by 6? (a) 4916 (b) 4914 (c) 4921 (d) 4911
35. 35.What is the smallest number you must add to 270 to make it divisible by 4? (a) 3 (b) 2 (c) 1
36. 36.Which of these numbers is divisible by 11? (a) 3372 (b) 3356 (c) 3366 (d) 3381
37. 37.What is the remainder when 186 is divided by 12? (a) 8 (b) 6 (c) 7 (d) 5
38. 38.What is the smallest number you must add to 208 to make it divisible by 3? (a) 2 (b) 1
39. 39.What is the remainder when 321 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
40. 40.What is the smallest number you must add to 587 to make it divisible by 9? (a) 7 (b) 6 (c) 2 (d) 8

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