Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 112 · 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) 753 (b) 756 (c) 757 (d) 761
2. 2.What is the smallest number you must add to 167 to make it divisible by 3? (a) 2 (b) 1
3. 3.What is the remainder when 229 is divided by 11? (a) 2 (b) 9 (c) 10 (d) 8
4. 4.Which of these numbers is divisible by 3? (a) 1883 (b) 1881 (c) 1879 (d) 1886
5. 5.What is the remainder when 45 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
6. 6.Which of these numbers is divisible by 9? (a) 2110 (b) 2122 (c) 2106 (d) 2098
7. 7.Which of these numbers is divisible by 6? (a) 2202 (b) 2209 (c) 2199 (d) 2207
8. 8.Which of these numbers is divisible by 9? (a) 6353 (b) 6354 (c) 6356 (d) 6364
9. 9.What is the units digit of 8^31? (a) 8 (b) 2 (c) 4 (d) 6
10. 10.What is the remainder when 605 is divided by 10? (a) 4 (b) 6 (c) 7 (d) 5
11. 11.Which of these numbers is divisible by 9? (a) 1270 (b) 1260 (c) 1261 (d) 1253
12. 12.What is the smallest number you must add to 301 to make it divisible by 6? (a) 1 (b) 4 (c) 3 (d) 5
13. 13.Which of these numbers is divisible by 3? (a) 1045 (b) 1052 (c) 1047 (d) 1048
14. 14.What is the smallest number you must add to 278 to make it divisible by 8? (a) 6 (b) 3 (c) 2 (d) 1
15. 15.What is the remainder when 226 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 0
16. 16.Which of these numbers is divisible by 11? (a) 749 (b) 733 (c) 745 (d) 737
17. 17.What is the units digit of 7^31? (a) 3 (b) 1 (c) 7 (d) 9
18. 18.What is the units digit of 2^47? (a) 4 (b) 8 (c) 6 (d) 2
19. 19.What is the units digit of 4^27? (a) 3 (b) 4 (c) 5 (d) 6
20. 20.What is the units digit of 8^35? (a) 8 (b) 6 (c) 4 (d) 2
21. 21.What is the remainder when 452 is divided by 7? (a) 3 (b) 6 (c) 4 (d) 5
22. 22.What is the units digit of 4^38? (a) 5 (b) 6 (c) 4 (d) 7
23. 23.What is the units digit of 2^52? (a) 2 (b) 6 (c) 4 (d) 8
24. 24.Which of these numbers is divisible by 11? (a) 1236 (b) 1228 (c) 1253 (d) 1232
25. 25.What is the units digit of 9^41? (a) 7 (b) 9 (c) 8 (d) 1
26. 26.Which of these numbers is divisible by 3? (a) 1237 (b) 1234 (c) 1241 (d) 1236
27. 27.What is the remainder when 545 is divided by 10? (a) 5 (b) 4 (c) 7 (d) 6
28. 28.What is the smallest number you must add to 689 to make it divisible by 10? (a) 3 (b) 9 (c) 2 (d) 1
29. 29.What is the remainder when 91 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
30. 30.What is the units digit of 3^56? (a) 1 (b) 9 (c) 3 (d) 7
31. 31.What is the smallest number you must add to 82 to make it divisible by 4? (a) 2 (b) 1 (c) 3
32. 32.Which of these numbers is divisible by 11? (a) 3128 (b) 3145 (c) 3135 (d) 3152
33. 33.What is the smallest number you must add to 298 to make it divisible by 4? (a) 2 (b) 1 (c) 3
34. 34.What is the remainder when 423 is divided by 12? (a) 3 (b) 2 (c) 4 (d) 9
35. 35.What is the remainder when 588 is divided by 11? (a) 7 (b) 6 (c) 5 (d) 4
36. 36.What is the remainder when 208 is divided by 7? (a) 2 (b) 4 (c) 6 (d) 5
37. 37.Which of these numbers is divisible by 6? (a) 1086 (b) 1091 (c) 1082 (d) 1095
38. 38.What is the units digit of 2^21? (a) 6 (b) 8 (c) 4 (d) 2
39. 39.Which of these numbers is divisible by 3? (a) 2342 (b) 2338 (c) 2344 (d) 2340
40. 40.What is the units digit of 3^47? (a) 3 (b) 9 (c) 1 (d) 7

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