Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 237 · 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 9? (a) 3837 (b) 3829 (c) 3825 (d) 3819
2. 2.Which of these numbers is divisible by 3? (a) 2069 (b) 2063 (c) 2064 (d) 2066
3. 3.What is the remainder when 732 is divided by 10? (a) 8 (b) 2 (c) 1 (d) 3
4. 4.Which of these numbers is divisible by 11? (a) 5615 (b) 5594 (c) 5606 (d) 5599
5. 5.What is the smallest number you must add to 94 to make it divisible by 8? (a) 3 (b) 1 (c) 6 (d) 2
6. 6.Which of these numbers is divisible by 3? (a) 2113 (b) 2112 (c) 2110 (d) 2116
7. 7.What is the remainder when 166 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
8. 8.What is the units digit of 2^37? (a) 8 (b) 4 (c) 2 (d) 6
9. 9.Which of these numbers is divisible by 4? (a) 332 (b) 338 (c) 331 (d) 334
10. 10.What is the units digit of 3^10? (a) 1 (b) 7 (c) 9 (d) 3
11. 11.Which of these numbers is divisible by 6? (a) 1232 (b) 1224 (c) 1226 (d) 1220
12. 12.What is the units digit of 3^38? (a) 9 (b) 7 (c) 3 (d) 1
13. 13.Which of these numbers is divisible by 6? (a) 759 (b) 747 (c) 750 (d) 751
14. 14.Which of these numbers is divisible by 6? (a) 3695 (b) 3700 (c) 3704 (d) 3696
15. 15.Which of these numbers is divisible by 3? (a) 823 (b) 819 (c) 817 (d) 821
16. 16.Which of these numbers is divisible by 4? (a) 3379 (b) 3381 (c) 3373 (d) 3376
17. 17.What is the remainder when 62 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
18. 18.What is the units digit of 3^11? (a) 9 (b) 1 (c) 7 (d) 3
19. 19.What is the units digit of 2^28? (a) 4 (b) 6 (c) 8 (d) 2
20. 20.What is the smallest number you must add to 378 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
21. 21.Which of these numbers is divisible by 3? (a) 2540 (b) 2534 (c) 2535 (d) 2537
22. 22.Which of these numbers is divisible by 8? (a) 6407 (b) 6408 (c) 6412 (d) 6418
23. 23.What is the units digit of 7^42? (a) 3 (b) 1 (c) 7 (d) 9
24. 24.Which of these numbers is divisible by 8? (a) 339 (b) 335 (c) 336 (d) 351
25. 25.What is the remainder when 679 is divided by 10? (a) 8 (b) 9 (c) 7 (d) 1
26. 26.What is the remainder when 195 is divided by 7? (a) 1 (b) 6 (c) 5 (d) 4
27. 27.Which of these numbers is divisible by 4? (a) 518 (b) 512 (c) 513 (d) 510
28. 28.Which of these numbers is divisible by 9? (a) 5318 (b) 5319 (c) 5320 (d) 5332
29. 29.What is the remainder when 113 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
30. 30.Which of these numbers is divisible by 3? (a) 221 (b) 216 (c) 218 (d) 214
31. 31.Which of these numbers is divisible by 3? (a) 2395 (b) 2398 (c) 2402 (d) 2397
32. 32.What is the smallest number you must add to 65 to make it divisible by 3? (a) 2 (b) 1
33. 33.What is the units digit of 2^17? (a) 2 (b) 4 (c) 6 (d) 8
34. 34.What is the smallest number you must add to 91 to make it divisible by 4? (a) 1 (b) 3 (c) 2
35. 35.What is the smallest number you must add to 495 to make it divisible by 8? (a) 7 (b) 2 (c) 3 (d) 1
36. 36.What is the smallest number you must add to 229 to make it divisible by 4? (a) 1 (b) 3 (c) 2
37. 37.What is the remainder when 650 is divided by 9? (a) 7 (b) 3 (c) 2 (d) 1
38. 38.What is the smallest number you must add to 377 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
39. 39.What is the smallest number you must add to 130 to make it divisible by 8? (a) 2 (b) 5 (c) 7 (d) 6
40. 40.What is the remainder when 584 is divided by 10? (a) 4 (b) 5 (c) 6 (d) 3

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