Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 453 · 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.What is the smallest number you must add to 237 to make it divisible by 9? (a) 6 (b) 5 (c) 3 (d) 7
2. 2.What is the remainder when 180 is divided by 7? (a) 2 (b) 5 (c) 6 (d) 4
3. 3.What is the units digit of 8^20? (a) 4 (b) 2 (c) 8 (d) 6
4. 4.Which of these numbers is divisible by 6? (a) 1523 (b) 1527 (c) 1531 (d) 1524
5. 5.Which of these numbers is divisible by 4? (a) 3080 (b) 3079 (c) 3083 (d) 3087
6. 6.What is the units digit of 2^31? (a) 4 (b) 8 (c) 6 (d) 2
7. 7.What is the remainder when 307 is divided by 7? (a) 1 (b) 6 (c) 4 (d) 5
8. 8.What is the remainder when 222 is divided by 7? (a) 4 (b) 2 (c) 6 (d) 5
9. 9.Which of these numbers is divisible by 6? (a) 3216 (b) 3227 (c) 3213 (d) 3221
10. 10.What is the smallest number you must add to 115 to make it divisible by 9? (a) 1 (b) 2 (c) 7 (d) 3
11. 11.Which of these numbers is divisible by 8? (a) 6832 (b) 6837 (c) 6825 (d) 6847
12. 12.What is the remainder when 607 is divided by 11? (a) 2 (b) 1 (c) 9 (d) 3
13. 13.What is the remainder when 282 is divided by 7? (a) 3 (b) 5 (c) 2 (d) 1
14. 14.Which of these numbers is divisible by 11? (a) 8771 (b) 8791 (c) 8778 (d) 8788
15. 15.What is the smallest number you must add to 501 to make it divisible by 10? (a) 8 (b) 9 (c) 1 (d) 7
16. 16.Which of these numbers is divisible by 4? (a) 1872 (b) 1874 (c) 1869 (d) 1877
17. 17.What is the smallest number you must add to 147 to make it divisible by 9? (a) 5 (b) 6 (c) 7 (d) 3
18. 18.Which of these numbers is divisible by 9? (a) 2586 (b) 2592 (c) 2599 (d) 2602
19. 19.What is the units digit of 9^51? (a) 9 (b) 1 (c) 8 (d) 7
20. 20.What is the remainder when 719 is divided by 9? (a) 7 (b) 8 (c) 1 (d) 6
21. 21.What is the smallest number you must add to 332 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 4
22. 22.What is the units digit of 3^13? (a) 3 (b) 9 (c) 7 (d) 1
23. 23.Which of these numbers is divisible by 3? (a) 2298 (b) 2297 (c) 2299 (d) 2302
24. 24.Which of these numbers is divisible by 11? (a) 6314 (b) 6320 (c) 6304 (d) 6331
25. 25.What is the units digit of 3^47? (a) 3 (b) 1 (c) 7 (d) 9
26. 26.Which of these numbers is divisible by 4? (a) 437 (b) 435 (c) 436 (d) 442
27. 27.What is the remainder when 335 is divided by 11? (a) 5 (b) 4 (c) 7 (d) 6
28. 28.Which of these numbers is divisible by 4? (a) 1033 (b) 1029 (c) 1027 (d) 1028
29. 29.Which of these numbers is divisible by 4? (a) 204 (b) 202 (c) 206 (d) 210
30. 30.What is the smallest number you must add to 164 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
31. 31.What is the smallest number you must add to 116 to make it divisible by 6? (a) 4 (b) 5 (c) 2 (d) 3
32. 32.Which of these numbers is divisible by 4? (a) 1359 (b) 1355 (c) 1363 (d) 1356
33. 33.What is the units digit of 3^28? (a) 9 (b) 3 (c) 1 (d) 7
34. 34.What is the smallest number you must add to 429 to make it divisible by 9? (a) 2 (b) 4 (c) 6 (d) 3
35. 35.Which of these numbers is divisible by 9? (a) 3599 (b) 3582 (c) 3584 (d) 3577
36. 36.What is the units digit of 3^45? (a) 9 (b) 7 (c) 1 (d) 3
37. 37.What is the units digit of 7^42? (a) 7 (b) 1 (c) 9 (d) 3
38. 38.What is the units digit of 4^25? (a) 6 (b) 4 (c) 5 (d) 3
39. 39.Which of these numbers is divisible by 11? (a) 5502 (b) 5497 (c) 5512 (d) 5500
40. 40.What is the remainder when 428 is divided by 6? (a) 2 (b) 3 (c) 4 (d) 1

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