Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 684 · 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 8^38? (a) 2 (b) 8 (c) 6 (d) 4
2. 2.What is the units digit of 2^59? (a) 2 (b) 4 (c) 8 (d) 6
3. 3.Which of these numbers is divisible by 6? (a) 518 (b) 512 (c) 510 (d) 507
4. 4.Which of these numbers is divisible by 6? (a) 3382 (b) 3384 (c) 3386 (d) 3394
5. 5.What is the smallest number you must add to 104 to make it divisible by 7? (a) 3 (b) 2 (c) 1 (d) 6
6. 6.Which of these numbers is divisible by 9? (a) 3383 (b) 3395 (c) 3385 (d) 3384
7. 7.Which of these numbers is divisible by 8? (a) 3960 (b) 3955 (c) 3969 (d) 3966
8. 8.What is the units digit of 7^32? (a) 3 (b) 1 (c) 9 (d) 7
9. 9.What is the smallest number you must add to 284 to make it divisible by 11? (a) 3 (b) 9 (c) 1 (d) 2
10. 10.What is the units digit of 3^22? (a) 7 (b) 3 (c) 1 (d) 9
11. 11.What is the units digit of 4^16? (a) 7 (b) 4 (c) 5 (d) 6
12. 12.Which of these numbers is divisible by 8? (a) 1262 (b) 1277 (c) 1267 (d) 1264
13. 13.Which of these numbers is divisible by 11? (a) 3729 (b) 3737 (c) 3742 (d) 3726
14. 14.What is the units digit of 3^36? (a) 9 (b) 1 (c) 7 (d) 3
15. 15.What is the units digit of 2^35? (a) 8 (b) 6 (c) 2 (d) 4
16. 16.What is the remainder when 43 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
17. 17.What is the smallest number you must add to 163 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
18. 18.Which of these numbers is divisible by 11? (a) 4899 (b) 4914 (c) 4921 (d) 4906
19. 19.What is the units digit of 2^28? (a) 2 (b) 6 (c) 4 (d) 8
20. 20.What is the smallest number you must add to 126 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
21. 21.What is the units digit of 9^33? (a) 7 (b) 1 (c) 9 (d) 8
22. 22.What is the smallest number you must add to 139 to make it divisible by 3? (a) 1 (b) 2
23. 23.What is the smallest number you must add to 318 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
24. 24.What is the smallest number you must add to 67 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 5
25. 25.What is the remainder when 294 is divided by 12? (a) 6 (b) 7 (c) 8 (d) 5
26. 26.Which of these numbers is divisible by 11? (a) 7501 (b) 7480 (c) 7484 (d) 7474
27. 27.What is the units digit of 4^51? (a) 3 (b) 4 (c) 5 (d) 6
28. 28.What is the smallest number you must add to 522 to make it divisible by 8? (a) 6 (b) 7 (c) 5 (d) 2
29. 29.What is the smallest number you must add to 248 to make it divisible by 11? (a) 4 (b) 5 (c) 6 (d) 7
30. 30.Which of these numbers is divisible by 9? (a) 4587 (b) 4597 (c) 4581 (d) 4577
31. 31.What is the units digit of 3^42? (a) 1 (b) 7 (c) 9 (d) 3
32. 32.What is the units digit of 8^51? (a) 6 (b) 4 (c) 2 (d) 8
33. 33.Which of these numbers is divisible by 8? (a) 3233 (b) 3225 (c) 3219 (d) 3224
34. 34.What is the smallest number you must add to 584 to make it divisible by 12? (a) 3 (b) 8 (c) 5 (d) 4
35. 35.What is the units digit of 2^44? (a) 8 (b) 4 (c) 2 (d) 6
36. 36.What is the smallest number you must add to 141 to make it divisible by 8? (a) 3 (b) 4 (c) 2 (d) 5
37. 37.What is the units digit of 9^17? (a) 8 (b) 7 (c) 9 (d) 1
38. 38.What is the remainder when 70 is divided by 3? (a) 2 (b) 0 (c) 1
39. 39.What is the remainder when 406 is divided by 8? (a) 7 (b) 2 (c) 5 (d) 6
40. 40.Which of these numbers is divisible by 3? (a) 284 (b) 286 (c) 289 (d) 285

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