Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 881 · 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.Which of these numbers is divisible by 11? (a) 6879 (b) 6892 (c) 6886 (d) 6900
2. 2.What is the units digit of 4^44? (a) 7 (b) 4 (c) 5 (d) 6
3. 3.What is the units digit of 4^47? (a) 4 (b) 3 (c) 6 (d) 5
4. 4.What is the units digit of 8^58? (a) 2 (b) 8 (c) 6 (d) 4
5. 5.What is the smallest number you must add to 771 to make it divisible by 10? (a) 1 (b) 7 (c) 8 (d) 9
6. 6.What is the smallest number you must add to 392 to make it divisible by 12? (a) 4 (b) 5 (c) 8 (d) 3
7. 7.Which of these numbers is divisible by 3? (a) 124 (b) 120 (c) 119 (d) 122
8. 8.What is the remainder when 461 is divided by 10? (a) 9 (b) 1 (c) 0 (d) 2
9. 9.Which of these numbers is divisible by 3? (a) 266 (b) 261 (c) 260 (d) 262
10. 10.What is the remainder when 256 is divided by 9? (a) 6 (b) 5 (c) 3 (d) 4
11. 11.What is the smallest number you must add to 593 to make it divisible by 9? (a) 8 (b) 2 (c) 3 (d) 1
12. 12.What is the smallest number you must add to 248 to make it divisible by 10? (a) 3 (b) 8 (c) 2 (d) 1
13. 13.Which of these numbers is divisible by 9? (a) 7030 (b) 7041 (c) 7053 (d) 7038
14. 14.What is the smallest number you must add to 465 to make it divisible by 10? (a) 7 (b) 6 (c) 5 (d) 4
15. 15.What is the units digit of 3^44? (a) 1 (b) 9 (c) 3 (d) 7
16. 16.Which of these numbers is divisible by 11? (a) 7875 (b) 7876 (c) 7880 (d) 7890
17. 17.Which of these numbers is divisible by 11? (a) 6517 (b) 6512 (c) 6525 (d) 6503
18. 18.What is the smallest number you must add to 651 to make it divisible by 12? (a) 9 (b) 10 (c) 8 (d) 3
19. 19.Which of these numbers is divisible by 8? (a) 5552 (b) 5563 (c) 5558 (d) 5545
20. 20.What is the smallest number you must add to 92 to make it divisible by 3? (a) 1 (b) 2
21. 21.What is the units digit of 7^50? (a) 3 (b) 7 (c) 1 (d) 9
22. 22.Which of these numbers is divisible by 3? (a) 281 (b) 274 (c) 278 (d) 276
23. 23.What is the units digit of 4^60? (a) 7 (b) 6 (c) 5 (d) 4
24. 24.Which of these numbers is divisible by 11? (a) 1397 (b) 1394 (c) 1399 (d) 1416
25. 25.What is the remainder when 509 is divided by 8? (a) 3 (b) 6 (c) 5 (d) 4
26. 26.Which of these numbers is divisible by 11? (a) 2844 (b) 2838 (c) 2829 (d) 2858
27. 27.What is the remainder when 74 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
28. 28.What is the units digit of 2^41? (a) 6 (b) 4 (c) 8 (d) 2
29. 29.Which of these numbers is divisible by 4? (a) 2937 (b) 2941 (c) 2936 (d) 2933
30. 30.What is the remainder when 322 is divided by 6? (a) 4 (b) 5 (c) 2 (d) 3
31. 31.What is the smallest number you must add to 417 to make it divisible by 8? (a) 7 (b) 1 (c) 6 (d) 5
32. 32.What is the remainder when 535 is divided by 10? (a) 6 (b) 5 (c) 7 (d) 4
33. 33.What is the remainder when 199 is divided by 3? (a) 1 (b) 0 (c) 2
34. 34.What is the remainder when 712 is divided by 12? (a) 5 (b) 4 (c) 3 (d) 8
35. 35.Which of these numbers is divisible by 11? (a) 911 (b) 891 (c) 892 (d) 883
36. 36.What is the units digit of 2^30? (a) 2 (b) 6 (c) 8 (d) 4
37. 37.What is the units digit of 7^27? (a) 7 (b) 1 (c) 3 (d) 9
38. 38.What is the units digit of 4^26? (a) 4 (b) 7 (c) 5 (d) 6
39. 39.What is the smallest number you must add to 73 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
40. 40.What is the smallest number you must add to 55 to make it divisible by 3? (a) 2 (b) 1

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