Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 862 · 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 2^56? (a) 4 (b) 2 (c) 8 (d) 6
2. 2.What is the smallest number you must add to 247 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
3. 3.What is the smallest number you must add to 286 to make it divisible by 10? (a) 3 (b) 6 (c) 4 (d) 5
4. 4.What is the units digit of 4^27? (a) 5 (b) 4 (c) 6 (d) 3
5. 5.What is the smallest number you must add to 378 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
6. 6.What is the smallest number you must add to 884 to make it divisible by 11? (a) 6 (b) 7 (c) 8 (d) 4
7. 7.What is the remainder when 460 is divided by 9? (a) 0 (b) 1 (c) 8 (d) 2
8. 8.What is the remainder when 407 is divided by 6? (a) 1 (b) 5 (c) 3 (d) 4
9. 9.What is the smallest number you must add to 494 to make it divisible by 7? (a) 5 (b) 3 (c) 4 (d) 2
10. 10.What is the smallest number you must add to 140 to make it divisible by 8? (a) 6 (b) 5 (c) 4 (d) 3
11. 11.What is the remainder when 439 is divided by 10? (a) 8 (b) 9 (c) 1 (d) 7
12. 12.Which of these numbers is divisible by 4? (a) 209 (b) 207 (c) 203 (d) 204
13. 13.What is the remainder when 341 is divided by 5? (a) 0 (b) 4 (c) 2 (d) 1
14. 14.What is the units digit of 9^53? (a) 9 (b) 8 (c) 7 (d) 1
15. 15.What is the smallest number you must add to 151 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
16. 16.What is the units digit of 8^11? (a) 8 (b) 6 (c) 2 (d) 4
17. 17.Which of these numbers is divisible by 11? (a) 8440 (b) 8436 (c) 8437 (d) 8455
18. 18.Which of these numbers is divisible by 9? (a) 2299 (b) 2315 (c) 2309 (d) 2304
19. 19.What is the remainder when 161 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
20. 20.What is the remainder when 551 is divided by 10? (a) 1 (b) 2 (c) 0 (d) 9
21. 21.What is the units digit of 7^52? (a) 7 (b) 3 (c) 1 (d) 9
22. 22.What is the smallest number you must add to 62 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
23. 23.What is the units digit of 3^24? (a) 3 (b) 1 (c) 9 (d) 7
24. 24.What is the smallest number you must add to 418 to make it divisible by 6? (a) 2 (b) 1 (c) 3 (d) 4
25. 25.What is the remainder when 173 is divided by 7? (a) 5 (b) 2 (c) 4 (d) 6
26. 26.Which of these numbers is divisible by 9? (a) 7805 (b) 7794 (c) 7799 (d) 7791
27. 27.What is the units digit of 7^58? (a) 9 (b) 7 (c) 3 (d) 1
28. 28.Which of these numbers is divisible by 6? (a) 4901 (b) 4902 (c) 4912 (d) 4906
29. 29.Which of these numbers is divisible by 6? (a) 4645 (b) 4650 (c) 4653 (d) 4659
30. 30.What is the units digit of 2^13? (a) 4 (b) 8 (c) 6 (d) 2
31. 31.Which of these numbers is divisible by 3? (a) 2098 (b) 2097 (c) 2101 (d) 2095
32. 32.What is the remainder when 315 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
33. 33.What is the remainder when 153 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
34. 34.Which of these numbers is divisible by 8? (a) 5209 (b) 5208 (c) 5221 (d) 5207
35. 35.What is the units digit of 3^54? (a) 3 (b) 1 (c) 9 (d) 7
36. 36.What is the remainder when 42 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
37. 37.What is the smallest number you must add to 251 to make it divisible by 8? (a) 3 (b) 5 (c) 4 (d) 6
38. 38.Which of these numbers is divisible by 3? (a) 2141 (b) 2147 (c) 2143 (d) 2142
39. 39.What is the remainder when 352 is divided by 12? (a) 4 (b) 5 (c) 3 (d) 8
40. 40.Which of these numbers is divisible by 9? (a) 4733 (b) 4735 (c) 4734 (d) 4750

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