Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 584 · 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) 5186 (b) 5184 (c) 5200 (d) 5177
2. 2.Which of these numbers is divisible by 9? (a) 5471 (b) 5488 (c) 5473 (d) 5472
3. 3.What is the smallest number you must add to 208 to make it divisible by 3? (a) 2 (b) 1
4. 4.What is the remainder when 77 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
5. 5.What is the remainder when 555 is divided by 9? (a) 6 (b) 3 (c) 7 (d) 5
6. 6.Which of these numbers is divisible by 9? (a) 3496 (b) 3483 (c) 3484 (d) 3476
7. 7.What is the units digit of 9^43? (a) 9 (b) 7 (c) 1 (d) 8
8. 8.What is the smallest number you must add to 294 to make it divisible by 4? (a) 3 (b) 2 (c) 1
9. 9.What is the smallest number you must add to 208 to make it divisible by 7? (a) 1 (b) 5 (c) 3 (d) 2
10. 10.Which of these numbers is divisible by 11? (a) 1557 (b) 1565 (c) 1545 (d) 1551
11. 11.What is the smallest number you must add to 252 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
12. 12.What is the remainder when 451 is divided by 7? (a) 3 (b) 4 (c) 5 (d) 2
13. 13.What is the remainder when 392 is divided by 6? (a) 4 (b) 2 (c) 1 (d) 3
14. 14.What is the units digit of 2^34? (a) 2 (b) 6 (c) 4 (d) 8
15. 15.What is the units digit of 7^47? (a) 9 (b) 7 (c) 1 (d) 3
16. 16.What is the units digit of 4^17? (a) 4 (b) 6 (c) 5 (d) 3
17. 17.What is the smallest number you must add to 214 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
18. 18.What is the units digit of 3^59? (a) 1 (b) 9 (c) 3 (d) 7
19. 19.Which of these numbers is divisible by 6? (a) 2345 (b) 2350 (c) 2353 (d) 2346
20. 20.What is the units digit of 9^56? (a) 2 (b) 9 (c) 1 (d) 0
21. 21.What is the remainder when 182 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
22. 22.What is the smallest number you must add to 175 to make it divisible by 4? (a) 2 (b) 3 (c) 1
23. 23.What is the smallest number you must add to 657 to make it divisible by 11? (a) 8 (b) 2 (c) 3 (d) 4
24. 24.Which of these numbers is divisible by 3? (a) 205 (b) 209 (c) 211 (d) 207
25. 25.What is the remainder when 105 is divided by 6? (a) 3 (b) 5 (c) 4 (d) 2
26. 26.What is the smallest number you must add to 241 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
27. 27.Which of these numbers is divisible by 8? (a) 3235 (b) 3249 (c) 3240 (d) 3244
28. 28.What is the smallest number you must add to 283 to make it divisible by 9? (a) 6 (b) 5 (c) 7 (d) 4
29. 29.What is the units digit of 9^42? (a) 0 (b) 1 (c) 2 (d) 9
30. 30.What is the smallest number you must add to 191 to make it divisible by 9? (a) 6 (b) 8 (c) 7 (d) 2
31. 31.Which of these numbers is divisible by 6? (a) 2567 (b) 2552 (c) 2556 (d) 2558
32. 32.What is the smallest number you must add to 307 to make it divisible by 4? (a) 1 (b) 2 (c) 3
33. 33.What is the remainder when 432 is divided by 11? (a) 4 (b) 3 (c) 2 (d) 8
34. 34.What is the remainder when 76 is divided by 6? (a) 5 (b) 2 (c) 4 (d) 3
35. 35.Which of these numbers is divisible by 8? (a) 4505 (b) 4522 (c) 4512 (d) 4513
36. 36.What is the smallest number you must add to 459 to make it divisible by 6? (a) 4 (b) 2 (c) 3 (d) 5
37. 37.What is the remainder when 124 is divided by 3? (a) 2 (b) 0 (c) 1
38. 38.What is the remainder when 68 is divided by 3? (a) 1 (b) 2 (c) 0
39. 39.What is the remainder when 164 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
40. 40.What is the remainder when 561 is divided by 7? (a) 2 (b) 0 (c) 1 (d) 6

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