Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 919 · 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 222 to make it divisible by 8? (a) 2 (b) 6 (c) 3 (d) 1
2. 2.What is the units digit of 3^55? (a) 7 (b) 1 (c) 3 (d) 9
3. 3.What is the smallest number you must add to 175 to make it divisible by 4? (a) 2 (b) 1 (c) 3
4. 4.What is the units digit of 9^31? (a) 8 (b) 9 (c) 1 (d) 7
5. 5.What is the units digit of 2^25? (a) 8 (b) 4 (c) 2 (d) 6
6. 6.What is the remainder when 121 is divided by 5? (a) 2 (b) 4 (c) 0 (d) 1
7. 7.What is the remainder when 437 is divided by 8? (a) 4 (b) 3 (c) 5 (d) 6
8. 8.What is the remainder when 169 is divided by 6? (a) 2 (b) 1 (c) 5 (d) 0
9. 9.What is the smallest number you must add to 83 to make it divisible by 4? (a) 3 (b) 1 (c) 2
10. 10.Which of these numbers is divisible by 4? (a) 2446 (b) 2441 (c) 2438 (d) 2440
11. 11.Which of these numbers is divisible by 6? (a) 2026 (b) 2037 (c) 2031 (d) 2028
12. 12.What is the remainder when 49 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
13. 13.What is the smallest number you must add to 502 to make it divisible by 7? (a) 2 (b) 5 (c) 3 (d) 1
14. 14.Which of these numbers is divisible by 8? (a) 6940 (b) 6933 (c) 6936 (d) 6945
15. 15.Which of these numbers is divisible by 9? (a) 1765 (b) 1787 (c) 1776 (d) 1773
16. 16.What is the units digit of 4^58? (a) 5 (b) 6 (c) 7 (d) 4
17. 17.What is the units digit of 3^16? (a) 1 (b) 7 (c) 3 (d) 9
18. 18.What is the units digit of 8^40? (a) 8 (b) 2 (c) 4 (d) 6
19. 19.Which of these numbers is divisible by 8? (a) 5979 (b) 5984 (c) 5993 (d) 5988
20. 20.What is the units digit of 8^52? (a) 2 (b) 6 (c) 4 (d) 8
21. 21.What is the units digit of 3^31? (a) 3 (b) 1 (c) 9 (d) 7
22. 22.What is the remainder when 227 is divided by 8? (a) 4 (b) 3 (c) 2 (d) 5
23. 23.Which of these numbers is divisible by 11? (a) 2267 (b) 2256 (c) 2255 (d) 2245
24. 24.Which of these numbers is divisible by 3? (a) 1063 (b) 1067 (c) 1062 (d) 1060
25. 25.What is the units digit of 9^53? (a) 8 (b) 9 (c) 1 (d) 7
26. 26.What is the remainder when 346 is divided by 5? (a) 4 (b) 0 (c) 1 (d) 2
27. 27.Which of these numbers is divisible by 4? (a) 1044 (b) 1046 (c) 1043 (d) 1049
28. 28.What is the units digit of 3^28? (a) 3 (b) 7 (c) 9 (d) 1
29. 29.What is the units digit of 2^15? (a) 2 (b) 6 (c) 8 (d) 4
30. 30.What is the units digit of 4^17? (a) 4 (b) 5 (c) 3 (d) 6
31. 31.What is the units digit of 8^21? (a) 8 (b) 6 (c) 2 (d) 4
32. 32.What is the units digit of 2^12? (a) 4 (b) 6 (c) 8 (d) 2
33. 33.What is the remainder when 102 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
34. 34.What is the smallest number you must add to 294 to make it divisible by 11? (a) 2 (b) 3 (c) 4 (d) 8
35. 35.Which of these numbers is divisible by 6? (a) 2065 (b) 2064 (c) 2059 (d) 2071
36. 36.What is the smallest number you must add to 455 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 1
37. 37.Which of these numbers is divisible by 4? (a) 2568 (b) 2565 (c) 2569 (d) 2573
38. 38.What is the units digit of 2^21? (a) 4 (b) 8 (c) 2 (d) 6
39. 39.What is the remainder when 63 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
40. 40.What is the smallest number you must add to 139 to make it divisible by 11? (a) 7 (b) 5 (c) 3 (d) 4

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