Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 329 · 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 9^43? (a) 9 (b) 7 (c) 1 (d) 8
2. 2.What is the units digit of 8^18? (a) 6 (b) 2 (c) 4 (d) 8
3. 3.What is the remainder when 134 is divided by 5? (a) 2 (b) 4 (c) 3 (d) 1
4. 4.What is the smallest number you must add to 490 to make it divisible by 8? (a) 6 (b) 7 (c) 5 (d) 2
5. 5.What is the remainder when 372 is divided by 10? (a) 3 (b) 2 (c) 1 (d) 8
6. 6.What is the smallest number you must add to 113 to make it divisible by 8? (a) 6 (b) 5 (c) 1 (d) 7
7. 7.Which of these numbers is divisible by 3? (a) 151 (b) 157 (c) 154 (d) 153
8. 8.What is the smallest number you must add to 284 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
9. 9.What is the units digit of 3^26? (a) 7 (b) 3 (c) 1 (d) 9
10. 10.What is the smallest number you must add to 800 to make it divisible by 11? (a) 8 (b) 2 (c) 4 (d) 3
11. 11.Which of these numbers is divisible by 9? (a) 3840 (b) 3847 (c) 3830 (d) 3834
12. 12.What is the smallest number you must add to 366 to make it divisible by 10? (a) 6 (b) 5 (c) 3 (d) 4
13. 13.What is the smallest number you must add to 617 to make it divisible by 8? (a) 7 (b) 5 (c) 1 (d) 6
14. 14.Which of these numbers is divisible by 6? (a) 2700 (b) 2696 (c) 2703 (d) 2710
15. 15.What is the smallest number you must add to 551 to make it divisible by 10? (a) 8 (b) 9 (c) 7 (d) 1
16. 16.What is the remainder when 414 is divided by 7? (a) 6 (b) 0 (c) 1 (d) 2
17. 17.What is the smallest number you must add to 76 to make it divisible by 3? (a) 1 (b) 2
18. 18.What is the remainder when 164 is divided by 5? (a) 2 (b) 4 (c) 3 (d) 1
19. 19.Which of these numbers is divisible by 3? (a) 2174 (b) 2179 (c) 2177 (d) 2175
20. 20.Which of these numbers is divisible by 11? (a) 3658 (b) 3641 (c) 3645 (d) 3638
21. 21.Which of these numbers is divisible by 4? (a) 1928 (b) 1935 (c) 1926 (d) 1930
22. 22.What is the remainder when 413 is divided by 6? (a) 1 (b) 3 (c) 4 (d) 5
23. 23.Which of these numbers is divisible by 9? (a) 1546 (b) 1551 (c) 1564 (d) 1548
24. 24.What is the units digit of 2^58? (a) 8 (b) 6 (c) 2 (d) 4
25. 25.What is the remainder when 388 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
26. 26.Which of these numbers is divisible by 11? (a) 6340 (b) 6336 (c) 6355 (d) 6334
27. 27.What is the units digit of 8^28? (a) 8 (b) 6 (c) 2 (d) 4
28. 28.Which of these numbers is divisible by 11? (a) 9284 (b) 9282 (c) 9298 (d) 9291
29. 29.What is the remainder when 198 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
30. 30.Which of these numbers is divisible by 4? (a) 1910 (b) 1908 (c) 1906 (d) 1914
31. 31.What is the remainder when 199 is divided by 7? (a) 3 (b) 4 (c) 5 (d) 2
32. 32.What is the remainder when 117 is divided by 7? (a) 4 (b) 2 (c) 5 (d) 6
33. 33.Which of these numbers is divisible by 3? (a) 145 (b) 141 (c) 143 (d) 139
34. 34.What is the units digit of 4^14? (a) 7 (b) 5 (c) 4 (d) 6
35. 35.Which of these numbers is divisible by 11? (a) 9359 (b) 9368 (c) 9349 (d) 9350
36. 36.Which of these numbers is divisible by 4? (a) 2257 (b) 2251 (c) 2252 (d) 2253
37. 37.Which of these numbers is divisible by 6? (a) 4579 (b) 4585 (c) 4578 (d) 4574
38. 38.What is the smallest number you must add to 509 to make it divisible by 9? (a) 5 (b) 6 (c) 4 (d) 3
39. 39.Which of these numbers is divisible by 3? (a) 1369 (b) 1372 (c) 1371 (d) 1375
40. 40.Which of these numbers is divisible by 4? (a) 1125 (b) 1124 (c) 1129 (d) 1122

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