Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 66 · 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 8^39? (a) 2 (b) 8 (c) 6 (d) 4
2. 2.What is the units digit of 3^17? (a) 7 (b) 9 (c) 1 (d) 3
3. 3.What is the smallest number you must add to 299 to make it divisible by 11? (a) 8 (b) 10 (c) 9 (d) 2
4. 4.What is the remainder when 609 is divided by 9? (a) 3 (b) 7 (c) 6 (d) 5
5. 5.What is the units digit of 2^34? (a) 4 (b) 6 (c) 8 (d) 2
6. 6.What is the remainder when 149 is divided by 3? (a) 0 (b) 1 (c) 2
7. 7.Which of these numbers is divisible by 9? (a) 3666 (b) 3675 (c) 3663 (d) 3660
8. 8.What is the smallest number you must add to 165 to make it divisible by 4? (a) 2 (b) 1 (c) 3
9. 9.What is the smallest number you must add to 56 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
10. 10.Which of these numbers is divisible by 3? (a) 1204 (b) 1201 (c) 1198 (d) 1200
11. 11.What is the remainder when 61 is divided by 3? (a) 0 (b) 1 (c) 2
12. 12.What is the units digit of 4^23? (a) 6 (b) 3 (c) 5 (d) 4
13. 13.What is the smallest number you must add to 621 to make it divisible by 12? (a) 3 (b) 4 (c) 9 (d) 2
14. 14.What is the units digit of 7^22? (a) 7 (b) 9 (c) 3 (d) 1
15. 15.What is the smallest number you must add to 634 to make it divisible by 10? (a) 4 (b) 5 (c) 7 (d) 6
16. 16.What is the remainder when 80 is divided by 3? (a) 2 (b) 1 (c) 0
17. 17.What is the units digit of 7^20? (a) 9 (b) 1 (c) 7 (d) 3
18. 18.Which of these numbers is divisible by 3? (a) 2586 (b) 2590 (c) 2588 (d) 2585
19. 19.What is the smallest number you must add to 178 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
20. 20.What is the smallest number you must add to 411 to make it divisible by 8? (a) 5 (b) 3 (c) 4 (d) 6
21. 21.What is the remainder when 418 is divided by 7? (a) 4 (b) 5 (c) 2 (d) 6
22. 22.What is the remainder when 324 is divided by 5? (a) 3 (b) 4 (c) 2 (d) 1
23. 23.What is the smallest number you must add to 130 to make it divisible by 4? (a) 1 (b) 2 (c) 3
24. 24.What is the units digit of 9^20? (a) 2 (b) 0 (c) 9 (d) 1
25. 25.Which of these numbers is divisible by 4? (a) 229 (b) 224 (c) 226 (d) 223
26. 26.What is the units digit of 4^26? (a) 5 (b) 7 (c) 6 (d) 4
27. 27.What is the smallest number you must add to 167 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
28. 28.What is the remainder when 215 is divided by 8? (a) 6 (b) 1 (c) 5 (d) 7
29. 29.What is the remainder when 58 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
30. 30.What is the smallest number you must add to 559 to make it divisible by 9? (a) 6 (b) 8 (c) 7 (d) 1
31. 31.Which of these numbers is divisible by 8? (a) 1327 (b) 1312 (c) 1318 (d) 1307
32. 32.What is the remainder when 377 is divided by 8? (a) 7 (b) 2 (c) 1 (d) 0
33. 33.What is the units digit of 8^23? (a) 2 (b) 4 (c) 8 (d) 6
34. 34.Which of these numbers is divisible by 9? (a) 4338 (b) 4340 (c) 4337 (d) 4355
35. 35.What is the units digit of 9^28? (a) 1 (b) 0 (c) 2 (d) 9
36. 36.What is the units digit of 9^46? (a) 1 (b) 0 (c) 2 (d) 9
37. 37.Which of these numbers is divisible by 4? (a) 2439 (b) 2442 (c) 2435 (d) 2436
38. 38.What is the units digit of 9^26? (a) 9 (b) 0 (c) 1 (d) 2
39. 39.What is the smallest number you must add to 192 to make it divisible by 7? (a) 3 (b) 4 (c) 6 (d) 5
40. 40.What is the units digit of 8^24? (a) 4 (b) 8 (c) 6 (d) 2

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