Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 336 · 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 smallest number you must add to 397 to make it divisible by 9? (a) 6 (b) 8 (c) 1 (d) 7
2. 2.Which of these numbers is divisible by 6? (a) 1818 (b) 1817 (c) 1827 (d) 1821
3. 3.What is the remainder when 185 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
4. 4.What is the smallest number you must add to 154 to make it divisible by 4? (a) 2 (b) 3 (c) 1
5. 5.Which of these numbers is divisible by 8? (a) 548 (b) 544 (c) 537 (d) 555
6. 6.What is the remainder when 137 is divided by 3? (a) 1 (b) 2 (c) 0
7. 7.What is the remainder when 102 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
8. 8.Which of these numbers is divisible by 8? (a) 2719 (b) 2710 (c) 2723 (d) 2712
9. 9.Which of these numbers is divisible by 11? (a) 3158 (b) 3147 (c) 3157 (d) 3173
10. 10.Which of these numbers is divisible by 4? (a) 3258 (b) 3256 (c) 3262 (d) 3254
11. 11.Which of these numbers is divisible by 9? (a) 4598 (b) 4579 (c) 4581 (d) 4585
12. 12.Which of these numbers is divisible by 11? (a) 3970 (b) 3959 (c) 3947 (d) 3949
13. 13.What is the units digit of 3^21? (a) 1 (b) 9 (c) 3 (d) 7
14. 14.What is the remainder when 304 is divided by 7? (a) 4 (b) 3 (c) 2 (d) 5
15. 15.What is the units digit of 9^30? (a) 1 (b) 9 (c) 2 (d) 0
16. 16.Which of these numbers is divisible by 4? (a) 2868 (b) 2873 (c) 2867 (d) 2869
17. 17.Which of these numbers is divisible by 11? (a) 4582 (b) 4566 (c) 4556 (d) 4565
18. 18.What is the remainder when 109 is divided by 6? (a) 2 (b) 5 (c) 0 (d) 1
19. 19.Which of these numbers is divisible by 6? (a) 3073 (b) 3079 (c) 3072 (d) 3071
20. 20.What is the units digit of 2^32? (a) 4 (b) 8 (c) 6 (d) 2
21. 21.Which of these numbers is divisible by 9? (a) 1153 (b) 1142 (c) 1143 (d) 1145
22. 22.What is the smallest number you must add to 73 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
23. 23.What is the units digit of 7^31? (a) 9 (b) 7 (c) 1 (d) 3
24. 24.Which of these numbers is divisible by 9? (a) 765 (b) 762 (c) 773 (d) 778
25. 25.Which of these numbers is divisible by 6? (a) 2390 (b) 2396 (c) 2384 (d) 2388
26. 26.Which of these numbers is divisible by 9? (a) 7927 (b) 7918 (c) 7906 (d) 7911
27. 27.What is the smallest number you must add to 80 to make it divisible by 3? (a) 2 (b) 1
28. 28.What is the smallest number you must add to 422 to make it divisible by 11? (a) 4 (b) 7 (c) 6 (d) 8
29. 29.Which of these numbers is divisible by 3? (a) 1693 (b) 1691 (c) 1688 (d) 1689
30. 30.What is the smallest number you must add to 143 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
31. 31.What is the smallest number you must add to 164 to make it divisible by 8? (a) 4 (b) 6 (c) 3 (d) 5
32. 32.What is the smallest number you must add to 206 to make it divisible by 8? (a) 1 (b) 6 (c) 3 (d) 2
33. 33.What is the units digit of 7^24? (a) 7 (b) 9 (c) 3 (d) 1
34. 34.Which of these numbers is divisible by 6? (a) 3944 (b) 3949 (c) 3939 (d) 3942
35. 35.Which of these numbers is divisible by 4? (a) 2325 (b) 2333 (c) 2328 (d) 2330
36. 36.What is the units digit of 4^11? (a) 5 (b) 3 (c) 6 (d) 4
37. 37.What is the units digit of 8^26? (a) 6 (b) 4 (c) 2 (d) 8
38. 38.Which of these numbers is divisible by 3? (a) 484 (b) 480 (c) 482 (d) 479
39. 39.What is the smallest number you must add to 399 to make it divisible by 12? (a) 9 (b) 10 (c) 3 (d) 8
40. 40.What is the remainder when 403 is divided by 8? (a) 2 (b) 4 (c) 3 (d) 5

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