Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 353 · 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 9^56? (a) 9 (b) 2 (c) 1 (d) 0
2. 2.What is the remainder when 339 is divided by 7? (a) 2 (b) 5 (c) 4 (d) 3
3. 3.What is the units digit of 2^41? (a) 6 (b) 8 (c) 2 (d) 4
4. 4.What is the units digit of 2^18? (a) 4 (b) 8 (c) 6 (d) 2
5. 5.Which of these numbers is divisible by 4? (a) 3323 (b) 3316 (c) 3318 (d) 3313
6. 6.Which of these numbers is divisible by 11? (a) 3441 (b) 3455 (c) 3446 (d) 3443
7. 7.What is the remainder when 188 is divided by 7? (a) 4 (b) 6 (c) 5 (d) 1
8. 8.What is the units digit of 8^47? (a) 6 (b) 2 (c) 4 (d) 8
9. 9.What is the smallest number you must add to 430 to make it divisible by 6? (a) 1 (b) 3 (c) 2 (d) 4
10. 10.What is the smallest number you must add to 205 to make it divisible by 9? (a) 2 (b) 3 (c) 1 (d) 7
11. 11.What is the units digit of 8^16? (a) 4 (b) 8 (c) 2 (d) 6
12. 12.What is the remainder when 212 is divided by 11? (a) 8 (b) 2 (c) 3 (d) 4
13. 13.What is the smallest number you must add to 601 to make it divisible by 8? (a) 7 (b) 1 (c) 6 (d) 5
14. 14.What is the remainder when 318 is divided by 7? (a) 3 (b) 4 (c) 2 (d) 5
15. 15.Which of these numbers is divisible by 3? (a) 1258 (b) 1256 (c) 1257 (d) 1261
16. 16.What is the units digit of 9^24? (a) 2 (b) 0 (c) 9 (d) 1
17. 17.What is the units digit of 9^30? (a) 9 (b) 2 (c) 1 (d) 0
18. 18.What is the smallest number you must add to 271 to make it divisible by 4? (a) 3 (b) 2 (c) 1
19. 19.What is the units digit of 4^12? (a) 4 (b) 7 (c) 5 (d) 6
20. 20.What is the units digit of 3^30? (a) 3 (b) 1 (c) 9 (d) 7
21. 21.What is the remainder when 99 is divided by 8? (a) 5 (b) 3 (c) 4 (d) 2
22. 22.Which of these numbers is divisible by 3? (a) 637 (b) 634 (c) 641 (d) 636
23. 23.Which of these numbers is divisible by 11? (a) 5621 (b) 5629 (c) 5614 (d) 5633
24. 24.What is the smallest number you must add to 169 to make it divisible by 8? (a) 5 (b) 6 (c) 7 (d) 1
25. 25.What is the remainder when 664 is divided by 9? (a) 8 (b) 7 (c) 6 (d) 2
26. 26.What is the units digit of 8^44? (a) 4 (b) 2 (c) 6 (d) 8
27. 27.What is the smallest number you must add to 129 to make it divisible by 8? (a) 6 (b) 5 (c) 1 (d) 7
28. 28.What is the units digit of 9^60? (a) 9 (b) 0 (c) 2 (d) 1
29. 29.What is the remainder when 269 is divided by 10? (a) 1 (b) 8 (c) 9 (d) 7
30. 30.What is the units digit of 2^38? (a) 6 (b) 8 (c) 4 (d) 2
31. 31.What is the remainder when 54 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
32. 32.Which of these numbers is divisible by 8? (a) 2233 (b) 2240 (c) 2251 (d) 2242
33. 33.What is the remainder when 269 is divided by 8? (a) 4 (b) 6 (c) 5 (d) 3
34. 34.What is the smallest number you must add to 194 to make it divisible by 4? (a) 1 (b) 3 (c) 2
35. 35.Which of these numbers is divisible by 11? (a) 3148 (b) 3164 (c) 3146 (d) 3142
36. 36.What is the units digit of 7^22? (a) 9 (b) 1 (c) 3 (d) 7
37. 37.What is the remainder when 537 is divided by 11? (a) 10 (b) 8 (c) 9 (d) 2
38. 38.What is the smallest number you must add to 174 to make it divisible by 8? (a) 6 (b) 2 (c) 1 (d) 3
39. 39.Which of these numbers is divisible by 4? (a) 2129 (b) 2128 (c) 2133 (d) 2126
40. 40.What is the units digit of 2^52? (a) 4 (b) 2 (c) 8 (d) 6

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