Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 936 · 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 71 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
2. 2.What is the smallest number you must add to 195 to make it divisible by 7? (a) 6 (b) 2 (c) 1 (d) 3
3. 3.Which of these numbers is divisible by 6? (a) 4301 (b) 4290 (c) 4287 (d) 4294
4. 4.Which of these numbers is divisible by 8? (a) 6373 (b) 6356 (c) 6360 (d) 6364
5. 5.Which of these numbers is divisible by 11? (a) 2389 (b) 2400 (c) 2398 (d) 2419
6. 6.What is the smallest number you must add to 158 to make it divisible by 6? (a) 3 (b) 5 (c) 4 (d) 2
7. 7.Which of these numbers is divisible by 3? (a) 2474 (b) 2475 (c) 2479 (d) 2476
8. 8.What is the units digit of 3^21? (a) 1 (b) 7 (c) 9 (d) 3
9. 9.What is the units digit of 3^60? (a) 7 (b) 1 (c) 9 (d) 3
10. 10.Which of these numbers is divisible by 11? (a) 8723 (b) 8721 (c) 8741 (d) 8725
11. 11.Which of these numbers is divisible by 6? (a) 3450 (b) 3455 (c) 3447 (d) 3458
12. 12.Which of these numbers is divisible by 6? (a) 2352 (b) 2356 (c) 2350 (d) 2359
13. 13.What is the remainder when 308 is divided by 8? (a) 6 (b) 4 (c) 5 (d) 3
14. 14.What is the smallest number you must add to 194 to make it divisible by 3? (a) 1 (b) 2
15. 15.What is the smallest number you must add to 111 to make it divisible by 10? (a) 8 (b) 1 (c) 7 (d) 9
16. 16.What is the units digit of 4^24? (a) 5 (b) 7 (c) 4 (d) 6
17. 17.Which of these numbers is divisible by 4? (a) 254 (b) 245 (c) 248 (d) 250
18. 18.What is the remainder when 561 is divided by 9? (a) 3 (b) 2 (c) 6 (d) 4
19. 19.What is the remainder when 206 is divided by 3? (a) 1 (b) 0 (c) 2
20. 20.What is the units digit of 4^55? (a) 3 (b) 4 (c) 6 (d) 5
21. 21.What is the smallest number you must add to 155 to make it divisible by 3? (a) 1 (b) 2
22. 22.What is the units digit of 9^38? (a) 9 (b) 2 (c) 1 (d) 0
23. 23.What is the smallest number you must add to 214 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
24. 24.Which of these numbers is divisible by 8? (a) 4223 (b) 4227 (c) 4224 (d) 4237
25. 25.Which of these numbers is divisible by 6? (a) 2630 (b) 2627 (c) 2636 (d) 2628
26. 26.What is the remainder when 343 is divided by 12? (a) 6 (b) 8 (c) 5 (d) 7
27. 27.What is the smallest number you must add to 156 to make it divisible by 10? (a) 6 (b) 5 (c) 3 (d) 4
28. 28.What is the remainder when 468 is divided by 8? (a) 3 (b) 6 (c) 4 (d) 5
29. 29.What is the smallest number you must add to 514 to make it divisible by 11? (a) 4 (b) 2 (c) 3 (d) 8
30. 30.What is the units digit of 2^31? (a) 6 (b) 2 (c) 4 (d) 8
31. 31.What is the remainder when 47 is divided by 3? (a) 2 (b) 0 (c) 1
32. 32.Which of these numbers is divisible by 8? (a) 1489 (b) 1477 (c) 1480 (d) 1485
33. 33.Which of these numbers is divisible by 6? (a) 5232 (b) 5231 (c) 5233 (d) 5239
34. 34.What is the remainder when 102 is divided by 10? (a) 8 (b) 1 (c) 3 (d) 2
35. 35.What is the units digit of 7^28? (a) 3 (b) 7 (c) 1 (d) 9
36. 36.Which of these numbers is divisible by 9? (a) 7271 (b) 7287 (c) 7280 (d) 7272
37. 37.Which of these numbers is divisible by 11? (a) 719 (b) 714 (c) 704 (d) 696
38. 38.Which of these numbers is divisible by 9? (a) 717 (b) 711 (c) 704 (d) 727
39. 39.Which of these numbers is divisible by 8? (a) 768 (b) 781 (c) 773 (d) 765
40. 40.Which of these numbers is divisible by 3? (a) 490 (b) 497 (c) 493 (d) 492

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