Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 204 · 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 7^25? (a) 7 (b) 1 (c) 3 (d) 9
2. 2.What is the remainder when 94 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
3. 3.What is the units digit of 8^43? (a) 6 (b) 2 (c) 4 (d) 8
4. 4.What is the units digit of 7^14? (a) 3 (b) 1 (c) 7 (d) 9
5. 5.Which of these numbers is divisible by 11? (a) 2294 (b) 2299 (c) 2309 (d) 2315
6. 6.What is the smallest number you must add to 105 to make it divisible by 4? (a) 2 (b) 3 (c) 1
7. 7.Which of these numbers is divisible by 9? (a) 1331 (b) 1314 (c) 1316 (d) 1311
8. 8.What is the remainder when 557 is divided by 7? (a) 4 (b) 6 (c) 5 (d) 3
9. 9.Which of these numbers is divisible by 6? (a) 4652 (b) 4660 (c) 4666 (d) 4656
10. 10.Which of these numbers is divisible by 8? (a) 1558 (b) 1571 (c) 1564 (d) 1560
11. 11.What is the smallest number you must add to 232 to make it divisible by 12? (a) 7 (b) 9 (c) 8 (d) 4
12. 12.What is the remainder when 682 is divided by 10? (a) 1 (b) 2 (c) 3 (d) 8
13. 13.What is the remainder when 72 is divided by 7? (a) 2 (b) 3 (c) 5 (d) 1
14. 14.What is the smallest number you must add to 506 to make it divisible by 7? (a) 2 (b) 6 (c) 4 (d) 5
15. 15.What is the units digit of 7^45? (a) 9 (b) 7 (c) 3 (d) 1
16. 16.What is the units digit of 9^54? (a) 0 (b) 9 (c) 2 (d) 1
17. 17.Which of these numbers is divisible by 4? (a) 2420 (b) 2426 (c) 2422 (d) 2417
18. 18.What is the units digit of 4^20? (a) 5 (b) 6 (c) 7 (d) 4
19. 19.What is the remainder when 268 is divided by 10? (a) 7 (b) 9 (c) 2 (d) 8
20. 20.Which of these numbers is divisible by 3? (a) 2289 (b) 2291 (c) 2288 (d) 2293
21. 21.What is the smallest number you must add to 99 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
22. 22.What is the units digit of 9^58? (a) 9 (b) 1 (c) 0 (d) 2
23. 23.What is the smallest number you must add to 199 to make it divisible by 4? (a) 1 (b) 2 (c) 3
24. 24.What is the units digit of 4^35? (a) 3 (b) 5 (c) 6 (d) 4
25. 25.What is the smallest number you must add to 226 to make it divisible by 7? (a) 5 (b) 4 (c) 6 (d) 2
26. 26.What is the units digit of 8^30? (a) 8 (b) 2 (c) 4 (d) 6
27. 27.What is the units digit of 7^26? (a) 7 (b) 3 (c) 9 (d) 1
28. 28.What is the remainder when 125 is divided by 3? (a) 1 (b) 2 (c) 0
29. 29.Which of these numbers is divisible by 4? (a) 2387 (b) 2383 (c) 2389 (d) 2384
30. 30.Which of these numbers is divisible by 6? (a) 3093 (b) 3089 (c) 3098 (d) 3090
31. 31.What is the smallest number you must add to 253 to make it divisible by 4? (a) 3 (b) 2 (c) 1
32. 32.What is the remainder when 138 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
33. 33.Which of these numbers is divisible by 8? (a) 4690 (b) 4686 (c) 4703 (d) 4688
34. 34.What is the units digit of 2^20? (a) 8 (b) 4 (c) 2 (d) 6
35. 35.What is the smallest number you must add to 189 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
36. 36.What is the units digit of 2^41? (a) 8 (b) 4 (c) 6 (d) 2
37. 37.What is the remainder when 346 is divided by 8? (a) 2 (b) 1 (c) 3 (d) 6
38. 38.What is the smallest number you must add to 93 to make it divisible by 6? (a) 5 (b) 2 (c) 4 (d) 3
39. 39.What is the remainder when 158 is divided by 7? (a) 3 (b) 6 (c) 4 (d) 5
40. 40.What is the smallest number you must add to 321 to make it divisible by 4? (a) 1 (b) 2 (c) 3

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