Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 437 · 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 340 to make it divisible by 7? (a) 4 (b) 2 (c) 3 (d) 5
2. 2.What is the units digit of 2^57? (a) 4 (b) 6 (c) 8 (d) 2
3. 3.Which of these numbers is divisible by 4? (a) 2808 (b) 2811 (c) 2805 (d) 2815
4. 4.What is the units digit of 3^19? (a) 3 (b) 1 (c) 9 (d) 7
5. 5.What is the units digit of 8^14? (a) 2 (b) 4 (c) 8 (d) 6
6. 6.What is the smallest number you must add to 122 to make it divisible by 4? (a) 3 (b) 2 (c) 1
7. 7.What is the smallest number you must add to 502 to make it divisible by 9? (a) 3 (b) 7 (c) 1 (d) 2
8. 8.Which of these numbers is divisible by 8? (a) 2654 (b) 2640 (c) 2645 (d) 2638
9. 9.Which of these numbers is divisible by 6? (a) 1146 (b) 1145 (c) 1154 (d) 1147
10. 10.Which of these numbers is divisible by 4? (a) 1857 (b) 1855 (c) 1856 (d) 1862
11. 11.Which of these numbers is divisible by 3? (a) 2453 (b) 2455 (c) 2454 (d) 2458
12. 12.Which of these numbers is divisible by 4? (a) 1013 (b) 1011 (c) 1008 (d) 1006
13. 13.Which of these numbers is divisible by 9? (a) 7793 (b) 7800 (c) 7785 (d) 7782
14. 14.What is the remainder when 323 is divided by 6? (a) 1 (b) 3 (c) 5 (d) 4
15. 15.Which of these numbers is divisible by 3? (a) 2064 (b) 2068 (c) 2063 (d) 2066
16. 16.What is the remainder when 166 is divided by 9? (a) 3 (b) 4 (c) 6 (d) 5
17. 17.Which of these numbers is divisible by 3? (a) 553 (b) 556 (c) 552 (d) 551
18. 18.Which of these numbers is divisible by 6? (a) 880 (b) 893 (c) 882 (d) 887
19. 19.What is the smallest number you must add to 44 to make it divisible by 3? (a) 2 (b) 1
20. 20.What is the remainder when 642 is divided by 8? (a) 1 (b) 6 (c) 3 (d) 2
21. 21.What is the remainder when 494 is divided by 7? (a) 3 (b) 6 (c) 5 (d) 4
22. 22.Which of these numbers is divisible by 8? (a) 469 (b) 456 (c) 449 (d) 459
23. 23.Which of these numbers is divisible by 11? (a) 3542 (b) 3541 (c) 3561 (d) 3551
24. 24.What is the remainder when 234 is divided by 5? (a) 4 (b) 1 (c) 3 (d) 2
25. 25.What is the remainder when 257 is divided by 7? (a) 6 (b) 4 (c) 5 (d) 2
26. 26.What is the remainder when 643 is divided by 8? (a) 2 (b) 3 (c) 5 (d) 4
27. 27.What is the remainder when 190 is divided by 9? (a) 8 (b) 2 (c) 0 (d) 1
28. 28.What is the smallest number you must add to 148 to make it divisible by 7? (a) 4 (b) 5 (c) 1 (d) 6
29. 29.What is the remainder when 749 is divided by 11? (a) 1 (b) 2 (c) 10 (d) 0
30. 30.What is the smallest number you must add to 140 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
31. 31.What is the units digit of 3^12? (a) 1 (b) 7 (c) 9 (d) 3
32. 32.What is the smallest number you must add to 138 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
33. 33.What is the units digit of 2^48? (a) 6 (b) 4 (c) 2 (d) 8
34. 34.What is the smallest number you must add to 481 to make it divisible by 7? (a) 1 (b) 5 (c) 3 (d) 2
35. 35.Which of these numbers is divisible by 4? (a) 372 (b) 371 (c) 377 (d) 374
36. 36.What is the smallest number you must add to 256 to make it divisible by 11? (a) 3 (b) 9 (c) 8 (d) 7
37. 37.Which of these numbers is divisible by 3? (a) 1256 (b) 1252 (c) 1259 (d) 1254
38. 38.What is the smallest number you must add to 181 to make it divisible by 9? (a) 7 (b) 6 (c) 8 (d) 1
39. 39.What is the units digit of 4^55? (a) 6 (b) 5 (c) 3 (d) 4
40. 40.Which of these numbers is divisible by 6? (a) 368 (b) 380 (c) 372 (d) 373

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