Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 828 · 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) 4399 (b) 4390 (c) 4392 (d) 4403
2. 2.What is the smallest number you must add to 292 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
3. 3.What is the remainder when 265 is divided by 6? (a) 2 (b) 5 (c) 0 (d) 1
4. 4.What is the remainder when 431 is divided by 6? (a) 4 (b) 3 (c) 5 (d) 1
5. 5.What is the smallest number you must add to 353 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
6. 6.What is the smallest number you must add to 201 to make it divisible by 4? (a) 2 (b) 1 (c) 3
7. 7.Which of these numbers is divisible by 3? (a) 1021 (b) 1028 (c) 1024 (d) 1023
8. 8.What is the units digit of 2^27? (a) 8 (b) 4 (c) 2 (d) 6
9. 9.Which of these numbers is divisible by 3? (a) 1793 (b) 1798 (c) 1794 (d) 1796
10. 10.What is the smallest number you must add to 502 to make it divisible by 9? (a) 7 (b) 3 (c) 1 (d) 2
11. 11.Which of these numbers is divisible by 8? (a) 5416 (b) 5429 (c) 5417 (d) 5414
12. 12.Which of these numbers is divisible by 3? (a) 1969 (b) 1975 (c) 1972 (d) 1971
13. 13.What is the smallest number you must add to 804 to make it divisible by 11? (a) 9 (b) 10 (c) 1 (d) 8
14. 14.What is the remainder when 623 is divided by 12? (a) 1 (b) 10 (c) 11 (d) 9
15. 15.What is the units digit of 4^11? (a) 5 (b) 6 (c) 3 (d) 4
16. 16.What is the units digit of 4^13? (a) 5 (b) 3 (c) 4 (d) 6
17. 17.What is the remainder when 91 is divided by 6? (a) 2 (b) 0 (c) 5 (d) 1
18. 18.Which of these numbers is divisible by 8? (a) 3837 (b) 3832 (c) 3846 (d) 3826
19. 19.What is the remainder when 184 is divided by 7? (a) 3 (b) 2 (c) 1 (d) 5
20. 20.What is the remainder when 73 is divided by 3? (a) 2 (b) 1 (c) 0
21. 21.Which of these numbers is divisible by 8? (a) 6848 (b) 6846 (c) 6858 (d) 6855
22. 22.What is the units digit of 7^46? (a) 7 (b) 3 (c) 9 (d) 1
23. 23.What is the remainder when 79 is divided by 3? (a) 1 (b) 0 (c) 2
24. 24.Which of these numbers is divisible by 11? (a) 6314 (b) 6306 (c) 6315 (d) 6327
25. 25.What is the smallest number you must add to 279 to make it divisible by 4? (a) 1 (b) 3 (c) 2
26. 26.Which of these numbers is divisible by 4? (a) 2474 (b) 2472 (c) 2469 (d) 2479
27. 27.What is the remainder when 65 is divided by 3? (a) 2 (b) 1 (c) 0
28. 28.What is the smallest number you must add to 271 to make it divisible by 10? (a) 9 (b) 8 (c) 7 (d) 1
29. 29.Which of these numbers is divisible by 11? (a) 6322 (b) 6325 (c) 6333 (d) 6344
30. 30.What is the remainder when 223 is divided by 9? (a) 7 (b) 6 (c) 2 (d) 8
31. 31.Which of these numbers is divisible by 4? (a) 236 (b) 241 (c) 233 (d) 238
32. 32.Which of these numbers is divisible by 3? (a) 1261 (b) 1265 (c) 1268 (d) 1263
33. 33.Which of these numbers is divisible by 9? (a) 8039 (b) 8058 (c) 8046 (d) 8052
34. 34.What is the remainder when 139 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
35. 35.What is the units digit of 2^30? (a) 4 (b) 2 (c) 6 (d) 8
36. 36.Which of these numbers is divisible by 6? (a) 4649 (b) 4639 (c) 4638 (d) 4634
37. 37.What is the remainder when 667 is divided by 10? (a) 8 (b) 6 (c) 7 (d) 3
38. 38.What is the smallest number you must add to 569 to make it divisible by 10? (a) 3 (b) 9 (c) 1 (d) 2
39. 39.What is the remainder when 117 is divided by 8? (a) 6 (b) 5 (c) 3 (d) 4
40. 40.Which of these numbers is divisible by 4? (a) 1538 (b) 1533 (c) 1536 (d) 1542

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