Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 534 · 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 3? (a) 2572 (b) 2579 (c) 2576 (d) 2574
2. 2.Which of these numbers is divisible by 8? (a) 1114 (b) 1099 (c) 1104 (d) 1105
3. 3.What is the remainder when 38 is divided by 3? (a) 1 (b) 2 (c) 0
4. 4.Which of these numbers is divisible by 9? (a) 6134 (b) 6114 (c) 6120 (d) 6128
5. 5.What is the units digit of 3^53? (a) 9 (b) 7 (c) 1 (d) 3
6. 6.Which of these numbers is divisible by 9? (a) 6767 (b) 6750 (c) 6758 (d) 6746
7. 7.Which of these numbers is divisible by 3? (a) 354 (b) 356 (c) 352 (d) 358
8. 8.Which of these numbers is divisible by 9? (a) 3465 (b) 3476 (c) 3463 (d) 3467
9. 9.What is the units digit of 7^11? (a) 1 (b) 7 (c) 3 (d) 9
10. 10.Which of these numbers is divisible by 6? (a) 1845 (b) 1839 (c) 1832 (d) 1836
11. 11.What is the smallest number you must add to 389 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
12. 12.What is the remainder when 207 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
13. 13.What is the smallest number you must add to 644 to make it divisible by 8? (a) 3 (b) 4 (c) 6 (d) 5
14. 14.What is the remainder when 778 is divided by 10? (a) 7 (b) 8 (c) 2 (d) 9
15. 15.What is the remainder when 121 is divided by 3? (a) 0 (b) 2 (c) 1
16. 16.What is the units digit of 8^38? (a) 4 (b) 8 (c) 2 (d) 6
17. 17.What is the units digit of 9^35? (a) 8 (b) 9 (c) 1 (d) 7
18. 18.What is the remainder when 209 is divided by 9? (a) 7 (b) 3 (c) 2 (d) 1
19. 19.Which of these numbers is divisible by 4? (a) 1261 (b) 1258 (c) 1256 (d) 1255
20. 20.Which of these numbers is divisible by 3? (a) 2374 (b) 2377 (c) 2373 (d) 2372
21. 21.What is the smallest number you must add to 346 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
22. 22.What is the remainder when 114 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
23. 23.Which of these numbers is divisible by 11? (a) 6432 (b) 6403 (c) 6413 (d) 6414
24. 24.Which of these numbers is divisible by 6? (a) 4776 (b) 4772 (c) 4780 (d) 4785
25. 25.What is the remainder when 537 is divided by 8? (a) 0 (b) 7 (c) 2 (d) 1
26. 26.Which of these numbers is divisible by 6? (a) 1337 (b) 1327 (c) 1326 (d) 1324
27. 27.What is the remainder when 95 is divided by 3? (a) 1 (b) 0 (c) 2
28. 28.What is the units digit of 9^32? (a) 0 (b) 2 (c) 9 (d) 1
29. 29.Which of these numbers is divisible by 4? (a) 1960 (b) 1963 (c) 1958 (d) 1967
30. 30.Which of these numbers is divisible by 4? (a) 312 (b) 309 (c) 318 (d) 314
31. 31.Which of these numbers is divisible by 11? (a) 3908 (b) 3897 (c) 3917 (d) 3905
32. 32.What is the units digit of 4^23? (a) 3 (b) 5 (c) 6 (d) 4
33. 33.What is the smallest number you must add to 397 to make it divisible by 9? (a) 7 (b) 8 (c) 1 (d) 6
34. 34.What is the smallest number you must add to 729 to make it divisible by 11? (a) 9 (b) 8 (c) 3 (d) 7
35. 35.What is the units digit of 3^36? (a) 9 (b) 1 (c) 3 (d) 7
36. 36.What is the smallest number you must add to 485 to make it divisible by 7? (a) 5 (b) 2 (c) 6 (d) 4
37. 37.What is the units digit of 9^25? (a) 8 (b) 1 (c) 7 (d) 9
38. 38.What is the smallest number you must add to 530 to make it divisible by 7? (a) 1 (b) 3 (c) 2 (d) 5
39. 39.What is the units digit of 4^27? (a) 5 (b) 4 (c) 3 (d) 6
40. 40.What is the remainder when 302 is divided by 11? (a) 7 (b) 4 (c) 5 (d) 6

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