Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 62 · 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 3^40? (a) 3 (b) 9 (c) 7 (d) 1
2. 2.What is the units digit of 9^36? (a) 1 (b) 0 (c) 9 (d) 2
3. 3.What is the units digit of 7^39? (a) 7 (b) 9 (c) 3 (d) 1
4. 4.What is the remainder when 117 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
5. 5.What is the units digit of 3^39? (a) 7 (b) 3 (c) 1 (d) 9
6. 6.What is the remainder when 291 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
7. 7.What is the smallest number you must add to 162 to make it divisible by 4? (a) 2 (b) 1 (c) 3
8. 8.What is the units digit of 2^11? (a) 2 (b) 6 (c) 4 (d) 8
9. 9.What is the smallest number you must add to 34 to make it divisible by 3? (a) 1 (b) 2
10. 10.What is the smallest number you must add to 127 to make it divisible by 3? (a) 2 (b) 1
11. 11.What is the remainder when 209 is divided by 3? (a) 0 (b) 1 (c) 2
12. 12.What is the smallest number you must add to 382 to make it divisible by 11? (a) 2 (b) 4 (c) 8 (d) 3
13. 13.What is the remainder when 399 is divided by 8? (a) 5 (b) 1 (c) 7 (d) 6
14. 14.Which of these numbers is divisible by 3? (a) 2566 (b) 2564 (c) 2562 (d) 2560
15. 15.Which of these numbers is divisible by 9? (a) 6900 (b) 6885 (c) 6887 (d) 6881
16. 16.Which of these numbers is divisible by 11? (a) 8877 (b) 8886 (c) 8894 (d) 8876
17. 17.Which of these numbers is divisible by 8? (a) 4370 (b) 4368 (c) 4367 (d) 4383
18. 18.What is the units digit of 9^49? (a) 9 (b) 8 (c) 1 (d) 7
19. 19.What is the remainder when 235 is divided by 7? (a) 6 (b) 5 (c) 4 (d) 3
20. 20.What is the smallest number you must add to 620 to make it divisible by 8? (a) 3 (b) 6 (c) 4 (d) 5
21. 21.Which of these numbers is divisible by 11? (a) 567 (b) 550 (c) 556 (d) 548
22. 22.Which of these numbers is divisible by 8? (a) 4633 (b) 4647 (c) 4650 (d) 4640
23. 23.What is the remainder when 170 is divided by 12? (a) 3 (b) 1 (c) 2 (d) 10
24. 24.What is the smallest number you must add to 409 to make it divisible by 10? (a) 2 (b) 1 (c) 3 (d) 9
25. 25.What is the units digit of 2^25? (a) 4 (b) 2 (c) 6 (d) 8
26. 26.What is the smallest number you must add to 202 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
27. 27.What is the remainder when 643 is divided by 10? (a) 2 (b) 7 (c) 4 (d) 3
28. 28.What is the units digit of 2^33? (a) 4 (b) 6 (c) 8 (d) 2
29. 29.What is the remainder when 688 is divided by 9? (a) 6 (b) 5 (c) 4 (d) 3
30. 30.What is the units digit of 9^60? (a) 9 (b) 2 (c) 0 (d) 1
31. 31.What is the units digit of 4^49? (a) 3 (b) 4 (c) 6 (d) 5
32. 32.Which of these numbers is divisible by 6? (a) 571 (b) 583 (c) 576 (d) 581
33. 33.What is the remainder when 619 is divided by 8? (a) 4 (b) 5 (c) 2 (d) 3
34. 34.What is the smallest number you must add to 252 to make it divisible by 10? (a) 7 (b) 2 (c) 8 (d) 9
35. 35.What is the units digit of 4^46? (a) 4 (b) 5 (c) 7 (d) 6
36. 36.Which of these numbers is divisible by 3? (a) 2290 (b) 2293 (c) 2287 (d) 2289
37. 37.Which of these numbers is divisible by 6? (a) 5339 (b) 5333 (c) 5334 (d) 5343
38. 38.What is the remainder when 409 is divided by 11? (a) 3 (b) 1 (c) 2 (d) 9
39. 39.What is the smallest number you must add to 906 to make it divisible by 12? (a) 7 (b) 5 (c) 6 (d) 8
40. 40.What is the smallest number you must add to 327 to make it divisible by 12? (a) 3 (b) 8 (c) 9 (d) 10

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