Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 115 · 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 68 to make it divisible by 3? (a) 1 (b) 2
2. 2.Which of these numbers is divisible by 8? (a) 2093 (b) 2075 (c) 2084 (d) 2080
3. 3.What is the units digit of 2^43? (a) 8 (b) 2 (c) 6 (d) 4
4. 4.What is the units digit of 7^58? (a) 1 (b) 7 (c) 3 (d) 9
5. 5.What is the remainder when 226 is divided by 3? (a) 0 (b) 2 (c) 1
6. 6.What is the units digit of 4^46? (a) 4 (b) 7 (c) 5 (d) 6
7. 7.Which of these numbers is divisible by 6? (a) 4140 (b) 4151 (c) 4138 (d) 4145
8. 8.Which of these numbers is divisible by 4? (a) 3101 (b) 3096 (c) 3097 (d) 3094
9. 9.Which of these numbers is divisible by 11? (a) 1076 (b) 1096 (c) 1087 (d) 1078
10. 10.What is the smallest number you must add to 97 to make it divisible by 3? (a) 2 (b) 1
11. 11.What is the smallest number you must add to 207 to make it divisible by 4? (a) 1 (b) 3 (c) 2
12. 12.What is the units digit of 7^17? (a) 7 (b) 1 (c) 3 (d) 9
13. 13.What is the smallest number you must add to 400 to make it divisible by 6? (a) 3 (b) 2 (c) 1 (d) 4
14. 14.What is the smallest number you must add to 506 to make it divisible by 12? (a) 2 (b) 11 (c) 9 (d) 10
15. 15.What is the units digit of 4^19? (a) 5 (b) 6 (c) 3 (d) 4
16. 16.What is the remainder when 279 is divided by 8? (a) 5 (b) 1 (c) 7 (d) 6
17. 17.What is the units digit of 4^31? (a) 3 (b) 5 (c) 4 (d) 6
18. 18.What is the remainder when 83 is divided by 7? (a) 1 (b) 5 (c) 4 (d) 6
19. 19.What is the remainder when 553 is divided by 8? (a) 0 (b) 7 (c) 1 (d) 2
20. 20.What is the smallest number you must add to 71 to make it divisible by 4? (a) 3 (b) 2 (c) 1
21. 21.What is the smallest number you must add to 159 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 4
22. 22.What is the remainder when 402 is divided by 10? (a) 8 (b) 1 (c) 2 (d) 3
23. 23.What is the units digit of 4^56? (a) 7 (b) 5 (c) 4 (d) 6
24. 24.What is the remainder when 65 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
25. 25.What is the units digit of 7^36? (a) 7 (b) 3 (c) 1 (d) 9
26. 26.Which of these numbers is divisible by 9? (a) 6826 (b) 6811 (c) 6813 (d) 6815
27. 27.What is the smallest number you must add to 121 to make it divisible by 4? (a) 3 (b) 2 (c) 1
28. 28.What is the units digit of 2^25? (a) 2 (b) 4 (c) 6 (d) 8
29. 29.What is the remainder when 276 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 0
30. 30.What is the remainder when 213 is divided by 11? (a) 3 (b) 5 (c) 4 (d) 7
31. 31.Which of these numbers is divisible by 3? (a) 571 (b) 575 (c) 568 (d) 570
32. 32.Which of these numbers is divisible by 9? (a) 3038 (b) 3059 (c) 3044 (d) 3042
33. 33.Which of these numbers is divisible by 11? (a) 1826 (b) 1839 (c) 1824 (d) 1833
34. 34.Which of these numbers is divisible by 9? (a) 6348 (b) 6336 (c) 6343 (d) 6330
35. 35.What is the smallest number you must add to 147 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
36. 36.Which of these numbers is divisible by 11? (a) 2219 (b) 2202 (c) 2211 (d) 2232
37. 37.What is the units digit of 4^34? (a) 5 (b) 7 (c) 4 (d) 6
38. 38.Which of these numbers is divisible by 8? (a) 920 (b) 931 (c) 918 (d) 923
39. 39.What is the remainder when 185 is divided by 3? (a) 0 (b) 1 (c) 2
40. 40.What is the units digit of 2^48? (a) 8 (b) 6 (c) 4 (d) 2

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