Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 72 · 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) 6173 (b) 6182 (c) 6186 (d) 6194
2. 2.What is the remainder when 321 is divided by 6? (a) 5 (b) 3 (c) 4 (d) 2
3. 3.What is the units digit of 2^30? (a) 2 (b) 4 (c) 6 (d) 8
4. 4.What is the remainder when 132 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
5. 5.What is the remainder when 86 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
6. 6.What is the units digit of 2^38? (a) 6 (b) 2 (c) 8 (d) 4
7. 7.What is the smallest number you must add to 318 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
8. 8.Which of these numbers is divisible by 11? (a) 3455 (b) 3481 (c) 3465 (d) 3466
9. 9.What is the units digit of 2^16? (a) 8 (b) 4 (c) 6 (d) 2
10. 10.Which of these numbers is divisible by 11? (a) 9600 (b) 9576 (c) 9586 (d) 9581
11. 11.Which of these numbers is divisible by 3? (a) 1286 (b) 1291 (c) 1287 (d) 1288
12. 12.What is the units digit of 8^18? (a) 4 (b) 6 (c) 8 (d) 2
13. 13.What is the remainder when 137 is divided by 12? (a) 7 (b) 6 (c) 4 (d) 5
14. 14.What is the units digit of 9^10? (a) 9 (b) 2 (c) 1 (d) 0
15. 15.What is the remainder when 200 is divided by 12? (a) 7 (b) 9 (c) 8 (d) 4
16. 16.Which of these numbers is divisible by 6? (a) 2486 (b) 2480 (c) 2495 (d) 2484
17. 17.What is the remainder when 35 is divided by 3? (a) 0 (b) 1 (c) 2
18. 18.What is the units digit of 4^23? (a) 3 (b) 6 (c) 5 (d) 4
19. 19.What is the units digit of 8^21? (a) 4 (b) 8 (c) 6 (d) 2
20. 20.What is the units digit of 9^50? (a) 9 (b) 0 (c) 1 (d) 2
21. 21.What is the remainder when 247 is divided by 7? (a) 5 (b) 3 (c) 1 (d) 2
22. 22.What is the smallest number you must add to 641 to make it divisible by 9? (a) 7 (b) 6 (c) 2 (d) 8
23. 23.What is the remainder when 660 is divided by 9? (a) 6 (b) 4 (c) 2 (d) 3
24. 24.What is the remainder when 659 is divided by 9? (a) 2 (b) 1 (c) 7 (d) 3
25. 25.What is the remainder when 149 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
26. 26.Which of these numbers is divisible by 6? (a) 5184 (b) 5182 (c) 5194 (d) 5185
27. 27.What is the smallest number you must add to 259 to make it divisible by 4? (a) 1 (b) 2 (c) 3
28. 28.What is the remainder when 175 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
29. 29.What is the smallest number you must add to 206 to make it divisible by 4? (a) 1 (b) 3 (c) 2
30. 30.Which of these numbers is divisible by 9? (a) 1011 (b) 992 (c) 1005 (d) 999
31. 31.What is the remainder when 319 is divided by 12? (a) 6 (b) 8 (c) 5 (d) 7
32. 32.What is the smallest number you must add to 236 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
33. 33.Which of these numbers is divisible by 3? (a) 1267 (b) 1262 (c) 1265 (d) 1263
34. 34.Which of these numbers is divisible by 3? (a) 1455 (b) 1454 (c) 1460 (d) 1456
35. 35.What is the units digit of 9^16? (a) 0 (b) 2 (c) 1 (d) 9
36. 36.Which of these numbers is divisible by 3? (a) 1296 (b) 1298 (c) 1301 (d) 1294
37. 37.What is the smallest number you must add to 91 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
38. 38.What is the units digit of 9^25? (a) 8 (b) 7 (c) 1 (d) 9
39. 39.Which of these numbers is divisible by 3? (a) 1770 (b) 1775 (c) 1772 (d) 1769
40. 40.Which of these numbers is divisible by 8? (a) 377 (b) 362 (c) 371 (d) 368

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