Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 505 · 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 4? (a) 2401 (b) 2411 (c) 2404 (d) 2407
2. 2.Which of these numbers is divisible by 4? (a) 1332 (b) 1331 (c) 1334 (d) 1337
3. 3.Which of these numbers is divisible by 4? (a) 655 (b) 651 (c) 647 (d) 648
4. 4.What is the remainder when 271 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 0
5. 5.What is the remainder when 59 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
6. 6.What is the smallest number you must add to 119 to make it divisible by 3? (a) 2 (b) 1
7. 7.Which of these numbers is divisible by 8? (a) 6296 (b) 6295 (c) 6300 (d) 6308
8. 8.What is the units digit of 7^54? (a) 9 (b) 7 (c) 1 (d) 3
9. 9.Which of these numbers is divisible by 6? (a) 2365 (b) 2361 (c) 2353 (d) 2358
10. 10.Which of these numbers is divisible by 4? (a) 1272 (b) 1270 (c) 1279 (d) 1274
11. 11.What is the smallest number you must add to 307 to make it divisible by 6? (a) 5 (b) 1 (c) 3 (d) 4
12. 12.What is the units digit of 7^40? (a) 9 (b) 7 (c) 3 (d) 1
13. 13.What is the remainder when 187 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
14. 14.What is the remainder when 266 is divided by 10? (a) 7 (b) 4 (c) 6 (d) 5
15. 15.What is the smallest number you must add to 241 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 1
16. 16.What is the remainder when 217 is divided by 3? (a) 1 (b) 0 (c) 2
17. 17.What is the remainder when 190 is divided by 3? (a) 2 (b) 0 (c) 1
18. 18.Which of these numbers is divisible by 9? (a) 5587 (b) 5589 (c) 5592 (d) 5602
19. 19.Which of these numbers is divisible by 3? (a) 314 (b) 310 (c) 307 (d) 309
20. 20.What is the smallest number you must add to 335 to make it divisible by 7? (a) 1 (b) 6 (c) 2 (d) 3
21. 21.What is the smallest number you must add to 519 to make it divisible by 8? (a) 1 (b) 7 (c) 2 (d) 3
22. 22.What is the units digit of 2^48? (a) 6 (b) 4 (c) 8 (d) 2
23. 23.What is the units digit of 7^53? (a) 1 (b) 7 (c) 3 (d) 9
24. 24.What is the units digit of 4^43? (a) 6 (b) 4 (c) 5 (d) 3
25. 25.Which of these numbers is divisible by 8? (a) 2076 (b) 2080 (c) 2092 (d) 2084
26. 26.Which of these numbers is divisible by 11? (a) 1529 (b) 1531 (c) 1523 (d) 1545
27. 27.Which of these numbers is divisible by 4? (a) 3394 (b) 3388 (c) 3391 (d) 3385
28. 28.What is the remainder when 193 is divided by 10? (a) 7 (b) 4 (c) 3 (d) 2
29. 29.Which of these numbers is divisible by 6? (a) 3790 (b) 3801 (c) 3792 (d) 3796
30. 30.Which of these numbers is divisible by 6? (a) 2043 (b) 2034 (c) 2031 (d) 2035
31. 31.What is the remainder when 155 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
32. 32.Which of these numbers is divisible by 6? (a) 4531 (b) 4527 (c) 4530 (d) 4537
33. 33.What is the smallest number you must add to 808 to make it divisible by 11? (a) 6 (b) 5 (c) 8 (d) 7
34. 34.Which of these numbers is divisible by 4? (a) 1939 (b) 1931 (c) 1932 (d) 1935
35. 35.What is the smallest number you must add to 191 to make it divisible by 3? (a) 2 (b) 1
36. 36.What is the smallest number you must add to 230 to make it divisible by 3? (a) 2 (b) 1
37. 37.What is the smallest number you must add to 232 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
38. 38.What is the units digit of 9^16? (a) 9 (b) 2 (c) 1 (d) 0
39. 39.What is the units digit of 9^13? (a) 7 (b) 9 (c) 8 (d) 1
40. 40.What is the smallest number you must add to 926 to make it divisible by 12? (a) 2 (b) 9 (c) 11 (d) 10

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