Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 379 · 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 11? (a) 9604 (b) 9626 (c) 9624 (d) 9614
2. 2.What is the remainder when 492 is divided by 8? (a) 3 (b) 5 (c) 4 (d) 6
3. 3.What is the smallest number you must add to 310 to make it divisible by 12? (a) 1 (b) 10 (c) 2 (d) 3
4. 4.Which of these numbers is divisible by 8? (a) 6361 (b) 6357 (c) 6360 (d) 6370
5. 5.What is the remainder when 186 is divided by 8? (a) 3 (b) 6 (c) 1 (d) 2
6. 6.What is the smallest number you must add to 104 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 4
7. 7.What is the remainder when 239 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4
8. 8.Which of these numbers is divisible by 9? (a) 2893 (b) 2885 (c) 2903 (d) 2889
9. 9.What is the smallest number you must add to 178 to make it divisible by 4? (a) 2 (b) 1 (c) 3
10. 10.What is the remainder when 194 is divided by 11? (a) 7 (b) 4 (c) 8 (d) 6
11. 11.Which of these numbers is divisible by 3? (a) 1058 (b) 1060 (c) 1063 (d) 1059
12. 12.Which of these numbers is divisible by 3? (a) 2275 (b) 2274 (c) 2273 (d) 2279
13. 13.What is the smallest number you must add to 593 to make it divisible by 9? (a) 8 (b) 2 (c) 1 (d) 3
14. 14.What is the units digit of 2^19? (a) 6 (b) 8 (c) 2 (d) 4
15. 15.What is the remainder when 261 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
16. 16.What is the units digit of 2^51? (a) 2 (b) 6 (c) 4 (d) 8
17. 17.What is the units digit of 7^14? (a) 3 (b) 9 (c) 7 (d) 1
18. 18.What is the remainder when 496 is divided by 7? (a) 5 (b) 6 (c) 1 (d) 4
19. 19.What is the units digit of 4^31? (a) 6 (b) 4 (c) 5 (d) 3
20. 20.Which of these numbers is divisible by 8? (a) 7192 (b) 7206 (c) 7199 (d) 7185
21. 21.What is the remainder when 885 is divided by 11? (a) 6 (b) 5 (c) 7 (d) 4
22. 22.What is the units digit of 3^42? (a) 3 (b) 7 (c) 9 (d) 1
23. 23.What is the remainder when 440 is divided by 6? (a) 2 (b) 4 (c) 1 (d) 3
24. 24.Which of these numbers is divisible by 4? (a) 1318 (b) 1323 (c) 1320 (d) 1327
25. 25.What is the remainder when 258 is divided by 11? (a) 7 (b) 5 (c) 6 (d) 4
26. 26.Which of these numbers is divisible by 9? (a) 7409 (b) 7422 (c) 7407 (d) 7402
27. 27.Which of these numbers is divisible by 8? (a) 5596 (b) 5603 (c) 5592 (d) 5585
28. 28.What is the smallest number you must add to 507 to make it divisible by 8? (a) 4 (b) 6 (c) 3 (d) 5
29. 29.What is the smallest number you must add to 578 to make it divisible by 9? (a) 8 (b) 6 (c) 2 (d) 7
30. 30.What is the remainder when 507 is divided by 8? (a) 5 (b) 4 (c) 2 (d) 3
31. 31.What is the units digit of 7^21? (a) 9 (b) 3 (c) 7 (d) 1
32. 32.What is the remainder when 118 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
33. 33.What is the remainder when 701 is divided by 11? (a) 9 (b) 7 (c) 3 (d) 8
34. 34.What is the smallest number you must add to 143 to make it divisible by 6? (a) 2 (b) 3 (c) 1 (d) 5
35. 35.What is the units digit of 3^20? (a) 7 (b) 1 (c) 3 (d) 9
36. 36.What is the remainder when 627 is divided by 8? (a) 4 (b) 2 (c) 5 (d) 3
37. 37.What is the units digit of 3^41? (a) 3 (b) 1 (c) 7 (d) 9
38. 38.Which of these numbers is divisible by 3? (a) 295 (b) 291 (c) 292 (d) 290
39. 39.What is the units digit of 8^50? (a) 8 (b) 2 (c) 6 (d) 4
40. 40.What is the units digit of 4^41? (a) 6 (b) 4 (c) 3 (d) 5

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