Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 389 · 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 units digit of 9^22? (a) 0 (b) 2 (c) 9 (d) 1
2. 2.What is the smallest number you must add to 203 to make it divisible by 4? (a) 1 (b) 2 (c) 3
3. 3.What is the remainder when 129 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4
4. 4.What is the remainder when 121 is divided by 3? (a) 0 (b) 2 (c) 1
5. 5.What is the units digit of 9^14? (a) 9 (b) 1 (c) 2 (d) 0
6. 6.What is the smallest number you must add to 668 to make it divisible by 10? (a) 3 (b) 2 (c) 1 (d) 8
7. 7.Which of these numbers is divisible by 6? (a) 2825 (b) 2830 (c) 2820 (d) 2815
8. 8.What is the remainder when 301 is divided by 10? (a) 0 (b) 1 (c) 9 (d) 2
9. 9.Which of these numbers is divisible by 9? (a) 5977 (b) 5985 (c) 5999 (d) 5987
10. 10.Which of these numbers is divisible by 11? (a) 6405 (b) 6392 (c) 6415 (d) 6402
11. 11.Which of these numbers is divisible by 6? (a) 318 (b) 321 (c) 314 (d) 325
12. 12.What is the remainder when 366 is divided by 10? (a) 4 (b) 5 (c) 6 (d) 7
13. 13.What is the smallest number you must add to 580 to make it divisible by 9? (a) 5 (b) 6 (c) 4 (d) 7
14. 14.What is the smallest number you must add to 119 to make it divisible by 4? (a) 3 (b) 2 (c) 1
15. 15.Which of these numbers is divisible by 4? (a) 1349 (b) 1347 (c) 1344 (d) 1341
16. 16.Which of these numbers is divisible by 4? (a) 2642 (b) 2646 (c) 2640 (d) 2638
17. 17.What is the remainder when 501 is divided by 11? (a) 5 (b) 8 (c) 7 (d) 6
18. 18.What is the remainder when 188 is divided by 6? (a) 1 (b) 3 (c) 4 (d) 2
19. 19.What is the units digit of 7^10? (a) 3 (b) 9 (c) 7 (d) 1
20. 20.Which of these numbers is divisible by 6? (a) 2183 (b) 2178 (c) 2176 (d) 2187
21. 21.What is the remainder when 577 is divided by 9? (a) 0 (b) 8 (c) 2 (d) 1
22. 22.What is the units digit of 8^10? (a) 4 (b) 2 (c) 8 (d) 6
23. 23.What is the smallest number you must add to 508 to make it divisible by 9? (a) 5 (b) 6 (c) 7 (d) 4
24. 24.What is the remainder when 436 is divided by 10? (a) 6 (b) 7 (c) 4 (d) 5
25. 25.What is the units digit of 8^33? (a) 6 (b) 4 (c) 8 (d) 2
26. 26.Which of these numbers is divisible by 8? (a) 6371 (b) 6363 (c) 6360 (d) 6357
27. 27.What is the units digit of 7^42? (a) 1 (b) 9 (c) 3 (d) 7
28. 28.What is the smallest number you must add to 209 to make it divisible by 7? (a) 1 (b) 3 (c) 6 (d) 2
29. 29.What is the units digit of 2^22? (a) 6 (b) 2 (c) 4 (d) 8
30. 30.What is the units digit of 7^57? (a) 3 (b) 9 (c) 1 (d) 7
31. 31.What is the smallest number you must add to 125 to make it divisible by 3? (a) 2 (b) 1
32. 32.What is the units digit of 4^11? (a) 6 (b) 5 (c) 4 (d) 3
33. 33.What is the smallest number you must add to 97 to make it divisible by 9? (a) 2 (b) 1 (c) 3 (d) 7
34. 34.What is the units digit of 8^52? (a) 2 (b) 4 (c) 8 (d) 6
35. 35.Which of these numbers is divisible by 6? (a) 4604 (b) 4613 (c) 4601 (d) 4602
36. 36.What is the remainder when 103 is divided by 3? (a) 0 (b) 1 (c) 2
37. 37.What is the remainder when 221 is divided by 5? (a) 2 (b) 4 (c) 0 (d) 1
38. 38.Which of these numbers is divisible by 3? (a) 2245 (b) 2249 (c) 2252 (d) 2247
39. 39.What is the units digit of 3^33? (a) 7 (b) 1 (c) 3 (d) 9
40. 40.What is the smallest number you must add to 323 to make it divisible by 10? (a) 7 (b) 3 (c) 8 (d) 6

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