Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 86 · 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 remainder when 234 is divided by 8? (a) 3 (b) 2 (c) 1 (d) 6
2. 2.Which of these numbers is divisible by 3? (a) 1339 (b) 1334 (c) 1337 (d) 1335
3. 3.Which of these numbers is divisible by 4? (a) 2458 (b) 2460 (c) 2461 (d) 2467
4. 4.What is the units digit of 2^40? (a) 2 (b) 4 (c) 8 (d) 6
5. 5.Which of these numbers is divisible by 11? (a) 4728 (b) 4719 (c) 4713 (d) 4734
6. 6.What is the remainder when 293 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
7. 7.Which of these numbers is divisible by 4? (a) 3174 (b) 3168 (c) 3167 (d) 3170
8. 8.Which of these numbers is divisible by 4? (a) 3584 (b) 3589 (c) 3583 (d) 3586
9. 9.Which of these numbers is divisible by 8? (a) 1597 (b) 1592 (c) 1587 (d) 1605
10. 10.Which of these numbers is divisible by 8? (a) 2959 (b) 2944 (c) 2945 (d) 2941
11. 11.What is the remainder when 406 is divided by 9? (a) 2 (b) 1 (c) 0 (d) 8
12. 12.What is the smallest number you must add to 52 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
13. 13.Which of these numbers is divisible by 11? (a) 3306 (b) 3297 (c) 3289 (d) 3286
14. 14.What is the units digit of 3^41? (a) 3 (b) 1 (c) 9 (d) 7
15. 15.What is the units digit of 7^21? (a) 9 (b) 7 (c) 3 (d) 1
16. 16.What is the units digit of 4^12? (a) 7 (b) 4 (c) 5 (d) 6
17. 17.What is the smallest number you must add to 281 to make it divisible by 4? (a) 3 (b) 2 (c) 1
18. 18.What is the remainder when 328 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
19. 19.What is the remainder when 89 is divided by 8? (a) 2 (b) 1 (c) 0 (d) 7
20. 20.What is the remainder when 241 is divided by 8? (a) 2 (b) 1 (c) 0 (d) 7
21. 21.What is the remainder when 149 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
22. 22.What is the smallest number you must add to 543 to make it divisible by 9? (a) 7 (b) 5 (c) 3 (d) 6
23. 23.Which of these numbers is divisible by 6? (a) 5282 (b) 5274 (c) 5278 (d) 5272
24. 24.What is the units digit of 4^52? (a) 5 (b) 6 (c) 4 (d) 7
25. 25.What is the units digit of 7^45? (a) 9 (b) 3 (c) 1 (d) 7
26. 26.Which of these numbers is divisible by 4? (a) 2307 (b) 2308 (c) 2313 (d) 2310
27. 27.What is the remainder when 155 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
28. 28.What is the smallest number you must add to 242 to make it divisible by 3? (a) 1 (b) 2
29. 29.What is the remainder when 265 is divided by 7? (a) 6 (b) 4 (c) 5 (d) 1
30. 30.What is the remainder when 37 is divided by 3? (a) 2 (b) 0 (c) 1
31. 31.Which of these numbers is divisible by 8? (a) 4503 (b) 4504 (c) 4509 (d) 4517
32. 32.Which of these numbers is divisible by 9? (a) 2788 (b) 2776 (c) 2781 (d) 2797
33. 33.What is the units digit of 2^58? (a) 8 (b) 4 (c) 6 (d) 2
34. 34.What is the smallest number you must add to 94 to make it divisible by 4? (a) 3 (b) 2 (c) 1
35. 35.What is the units digit of 8^11? (a) 2 (b) 8 (c) 6 (d) 4
36. 36.What is the remainder when 261 is divided by 12? (a) 10 (b) 3 (c) 8 (d) 9
37. 37.What is the remainder when 313 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
38. 38.Which of these numbers is divisible by 4? (a) 1191 (b) 1194 (c) 1192 (d) 1197
39. 39.What is the remainder when 232 is divided by 3? (a) 0 (b) 1 (c) 2
40. 40.What is the smallest number you must add to 239 to make it divisible by 8? (a) 7 (b) 1 (c) 3 (d) 2

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