Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 849 · 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 4? (a) 226 (b) 229 (c) 234 (d) 228
2. 2.What is the remainder when 668 is divided by 10? (a) 2 (b) 7 (c) 8 (d) 9
3. 3.What is the smallest number you must add to 112 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
4. 4.What is the smallest number you must add to 658 to make it divisible by 10? (a) 2 (b) 3 (c) 1 (d) 8
5. 5.Which of these numbers is divisible by 3? (a) 1088 (b) 1086 (c) 1085 (d) 1090
6. 6.What is the remainder when 64 is divided by 6? (a) 5 (b) 4 (c) 3 (d) 2
7. 7.What is the remainder when 643 is divided by 9? (a) 6 (b) 5 (c) 4 (d) 3
8. 8.What is the smallest number you must add to 277 to make it divisible by 10? (a) 7 (b) 4 (c) 3 (d) 2
9. 9.What is the smallest number you must add to 541 to make it divisible by 8? (a) 4 (b) 5 (c) 3 (d) 2
10. 10.What is the units digit of 7^32? (a) 3 (b) 9 (c) 7 (d) 1
11. 11.What is the remainder when 76 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 0
12. 12.What is the remainder when 459 is divided by 10? (a) 7 (b) 8 (c) 9 (d) 1
13. 13.What is the smallest number you must add to 674 to make it divisible by 11? (a) 8 (b) 3 (c) 9 (d) 7
14. 14.What is the remainder when 484 is divided by 7? (a) 2 (b) 6 (c) 1 (d) 0
15. 15.What is the smallest number you must add to 245 to make it divisible by 12? (a) 5 (b) 8 (c) 7 (d) 6
16. 16.What is the smallest number you must add to 616 to make it divisible by 12? (a) 7 (b) 9 (c) 4 (d) 8
17. 17.What is the smallest number you must add to 277 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
18. 18.What is the smallest number you must add to 86 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
19. 19.What is the units digit of 2^45? (a) 8 (b) 6 (c) 2 (d) 4
20. 20.What is the units digit of 4^49? (a) 5 (b) 6 (c) 4 (d) 3
21. 21.Which of these numbers is divisible by 4? (a) 214 (b) 212 (c) 210 (d) 217
22. 22.What is the smallest number you must add to 654 to make it divisible by 10? (a) 6 (b) 7 (c) 5 (d) 4
23. 23.What is the smallest number you must add to 833 to make it divisible by 11? (a) 8 (b) 4 (c) 3 (d) 2
24. 24.What is the remainder when 142 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
25. 25.What is the remainder when 447 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
26. 26.What is the units digit of 9^56? (a) 9 (b) 2 (c) 1 (d) 0
27. 27.What is the units digit of 7^42? (a) 1 (b) 7 (c) 9 (d) 3
28. 28.Which of these numbers is divisible by 8? (a) 1983 (b) 1968 (c) 1963 (d) 1972
29. 29.Which of these numbers is divisible by 6? (a) 806 (b) 796 (c) 802 (d) 798
30. 30.What is the units digit of 2^28? (a) 6 (b) 8 (c) 2 (d) 4
31. 31.What is the units digit of 3^22? (a) 3 (b) 9 (c) 7 (d) 1
32. 32.What is the units digit of 8^14? (a) 2 (b) 6 (c) 8 (d) 4
33. 33.Which of these numbers is divisible by 9? (a) 1901 (b) 1923 (c) 1908 (d) 1912
34. 34.What is the remainder when 295 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
35. 35.Which of these numbers is divisible by 9? (a) 384 (b) 387 (c) 399 (d) 392
36. 36.What is the remainder when 179 is divided by 8? (a) 4 (b) 2 (c) 3 (d) 5
37. 37.What is the remainder when 62 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
38. 38.What is the remainder when 116 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 0
39. 39.Which of these numbers is divisible by 4? (a) 3314 (b) 3312 (c) 3317 (d) 3310
40. 40.Which of these numbers is divisible by 8? (a) 4691 (b) 4696 (c) 4711 (d) 4701

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