Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 696 · 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.Which of these numbers is divisible by 6? (a) 903 (b) 898 (c) 894 (d) 893
2. 2.What is the smallest number you must add to 147 to make it divisible by 4? (a) 3 (b) 1 (c) 2
3. 3.What is the units digit of 4^18? (a) 4 (b) 5 (c) 6 (d) 7
4. 4.What is the smallest number you must add to 68 to make it divisible by 6? (a) 3 (b) 2 (c) 5 (d) 4
5. 5.Which of these numbers is divisible by 9? (a) 6120 (b) 6116 (c) 6128 (d) 6133
6. 6.What is the remainder when 512 is divided by 7? (a) 2 (b) 6 (c) 1 (d) 0
7. 7.What is the units digit of 4^36? (a) 4 (b) 7 (c) 5 (d) 6
8. 8.What is the remainder when 323 is divided by 7? (a) 2 (b) 0 (c) 6 (d) 1
9. 9.What is the units digit of 3^37? (a) 7 (b) 3 (c) 9 (d) 1
10. 10.What is the smallest number you must add to 348 to make it divisible by 8? (a) 3 (b) 5 (c) 6 (d) 4
11. 11.What is the smallest number you must add to 111 to make it divisible by 4? (a) 1 (b) 2 (c) 3
12. 12.What is the units digit of 3^33? (a) 9 (b) 7 (c) 1 (d) 3
13. 13.Which of these numbers is divisible by 3? (a) 407 (b) 401 (c) 404 (d) 402
14. 14.What is the remainder when 458 is divided by 10? (a) 7 (b) 9 (c) 8 (d) 2
15. 15.What is the remainder when 106 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
16. 16.Which of these numbers is divisible by 3? (a) 229 (b) 225 (c) 223 (d) 226
17. 17.Which of these numbers is divisible by 3? (a) 855 (b) 860 (c) 854 (d) 856
18. 18.Which of these numbers is divisible by 11? (a) 6788 (b) 6798 (c) 6818 (d) 6801
19. 19.What is the smallest number you must add to 170 to make it divisible by 8? (a) 2 (b) 7 (c) 6 (d) 5
20. 20.What is the smallest number you must add to 502 to make it divisible by 7? (a) 3 (b) 2 (c) 5 (d) 1
21. 21.What is the remainder when 719 is divided by 12? (a) 10 (b) 1 (c) 9 (d) 11
22. 22.What is the units digit of 9^26? (a) 2 (b) 0 (c) 9 (d) 1
23. 23.Which of these numbers is divisible by 3? (a) 2339 (b) 2341 (c) 2335 (d) 2337
24. 24.What is the remainder when 353 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
25. 25.What is the remainder when 549 is divided by 10? (a) 8 (b) 1 (c) 9 (d) 7
26. 26.What is the units digit of 8^60? (a) 8 (b) 2 (c) 6 (d) 4
27. 27.What is the units digit of 3^17? (a) 1 (b) 3 (c) 9 (d) 7
28. 28.What is the smallest number you must add to 387 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
29. 29.Which of these numbers is divisible by 8? (a) 2431 (b) 2420 (c) 2436 (d) 2424
30. 30.Which of these numbers is divisible by 6? (a) 2818 (b) 2807 (c) 2812 (d) 2808
31. 31.Which of these numbers is divisible by 4? (a) 2930 (b) 2925 (c) 2933 (d) 2928
32. 32.What is the units digit of 2^27? (a) 2 (b) 8 (c) 4 (d) 6
33. 33.What is the remainder when 198 is divided by 7? (a) 2 (b) 5 (c) 1 (d) 3
34. 34.Which of these numbers is divisible by 4? (a) 3505 (b) 3509 (c) 3501 (d) 3504
35. 35.What is the remainder when 631 is divided by 10? (a) 2 (b) 9 (c) 1 (d) 0
36. 36.What is the remainder when 383 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
37. 37.What is the smallest number you must add to 85 to make it divisible by 7? (a) 1 (b) 5 (c) 4 (d) 6
38. 38.What is the smallest number you must add to 301 to make it divisible by 4? (a) 3 (b) 2 (c) 1
39. 39.What is the remainder when 159 is divided by 9? (a) 7 (b) 5 (c) 6 (d) 3
40. 40.Which of these numbers is divisible by 4? (a) 435 (b) 425 (c) 428 (d) 430

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