Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 381 · 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 units digit of 9^40? (a) 0 (b) 1 (c) 9 (d) 2
2. 2.What is the units digit of 2^11? (a) 6 (b) 8 (c) 4 (d) 2
3. 3.What is the smallest number you must add to 153 to make it divisible by 12? (a) 3 (b) 9 (c) 4 (d) 2
4. 4.What is the smallest number you must add to 421 to make it divisible by 7? (a) 4 (b) 6 (c) 1 (d) 5
5. 5.What is the units digit of 2^33? (a) 6 (b) 2 (c) 4 (d) 8
6. 6.Which of these numbers is divisible by 11? (a) 3135 (b) 3138 (c) 3127 (d) 3147
7. 7.What is the remainder when 159 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
8. 8.What is the remainder when 110 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
9. 9.Which of these numbers is divisible by 8? (a) 1961 (b) 1980 (c) 1974 (d) 1968
10. 10.What is the units digit of 7^20? (a) 3 (b) 9 (c) 1 (d) 7
11. 11.Which of these numbers is divisible by 4? (a) 3281 (b) 3291 (c) 3284 (d) 3286
12. 12.Which of these numbers is divisible by 3? (a) 295 (b) 293 (c) 298 (d) 294
13. 13.Which of these numbers is divisible by 3? (a) 2198 (b) 2195 (c) 2200 (d) 2196
14. 14.What is the remainder when 350 is divided by 6? (a) 4 (b) 1 (c) 2 (d) 3
15. 15.What is the smallest number you must add to 834 to make it divisible by 11? (a) 3 (b) 2 (c) 1 (d) 9
16. 16.What is the remainder when 344 is divided by 7? (a) 2 (b) 0 (c) 1 (d) 6
17. 17.Which of these numbers is divisible by 3? (a) 560 (b) 563 (c) 561 (d) 565
18. 18.Which of these numbers is divisible by 11? (a) 8084 (b) 8062 (c) 8063 (d) 8065
19. 19.What is the smallest number you must add to 106 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
20. 20.What is the remainder when 662 is divided by 9? (a) 5 (b) 7 (c) 6 (d) 4
21. 21.Which of these numbers is divisible by 6? (a) 4265 (b) 4255 (c) 4269 (d) 4260
22. 22.What is the remainder when 223 is divided by 6? (a) 2 (b) 5 (c) 0 (d) 1
23. 23.Which of these numbers is divisible by 8? (a) 667 (b) 678 (c) 658 (d) 664
24. 24.What is the remainder when 347 is divided by 12? (a) 11 (b) 1 (c) 10 (d) 9
25. 25.What is the remainder when 43 is divided by 3? (a) 2 (b) 1 (c) 0
26. 26.Which of these numbers is divisible by 11? (a) 4741 (b) 4734 (c) 4760 (d) 4749
27. 27.Which of these numbers is divisible by 3? (a) 2260 (b) 2255 (c) 2256 (d) 2257
28. 28.What is the remainder when 269 is divided by 10? (a) 8 (b) 9 (c) 7 (d) 1
29. 29.What is the units digit of 7^49? (a) 9 (b) 1 (c) 3 (d) 7
30. 30.What is the smallest number you must add to 192 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
31. 31.What is the smallest number you must add to 239 to make it divisible by 3? (a) 2 (b) 1
32. 32.Which of these numbers is divisible by 3? (a) 1339 (b) 1333 (c) 1337 (d) 1335
33. 33.What is the remainder when 700 is divided by 9? (a) 8 (b) 7 (c) 2 (d) 6
34. 34.What is the units digit of 2^52? (a) 6 (b) 8 (c) 2 (d) 4
35. 35.Which of these numbers is divisible by 6? (a) 693 (b) 704 (c) 696 (d) 701
36. 36.What is the smallest number you must add to 173 to make it divisible by 4? (a) 2 (b) 1 (c) 3
37. 37.What is the remainder when 591 is divided by 9? (a) 5 (b) 6 (c) 3 (d) 7
38. 38.What is the units digit of 3^35? (a) 3 (b) 1 (c) 7 (d) 9
39. 39.What is the remainder when 383 is divided by 11? (a) 2 (b) 8 (c) 10 (d) 9
40. 40.What is the smallest number you must add to 156 to make it divisible by 8? (a) 4 (b) 6 (c) 5 (d) 3

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