Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 56 · 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 6? (a) 3614 (b) 3618 (c) 3622 (d) 3627
2. 2.What is the remainder when 254 is divided by 6? (a) 1 (b) 3 (c) 4 (d) 2
3. 3.Which of these numbers is divisible by 4? (a) 608 (b) 613 (c) 607 (d) 610
4. 4.What is the units digit of 3^60? (a) 1 (b) 7 (c) 9 (d) 3
5. 5.What is the units digit of 3^51? (a) 9 (b) 1 (c) 3 (d) 7
6. 6.Which of these numbers is divisible by 11? (a) 2222 (b) 2232 (c) 2218 (d) 2240
7. 7.Which of these numbers is divisible by 4? (a) 1991 (b) 1986 (c) 1981 (d) 1984
8. 8.What is the smallest number you must add to 118 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
9. 9.What is the smallest number you must add to 529 to make it divisible by 11? (a) 9 (b) 10 (c) 1 (d) 8
10. 10.What is the remainder when 114 is divided by 8? (a) 2 (b) 3 (c) 6 (d) 1
11. 11.What is the units digit of 8^32? (a) 4 (b) 6 (c) 2 (d) 8
12. 12.What is the remainder when 578 is divided by 9? (a) 3 (b) 2 (c) 1 (d) 7
13. 13.Which of these numbers is divisible by 3? (a) 1879 (b) 1881 (c) 1885 (d) 1883
14. 14.What is the units digit of 9^24? (a) 9 (b) 2 (c) 0 (d) 1
15. 15.What is the smallest number you must add to 73 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
16. 16.What is the smallest number you must add to 500 to make it divisible by 9? (a) 3 (b) 6 (c) 5 (d) 4
17. 17.What is the smallest number you must add to 556 to make it divisible by 8? (a) 6 (b) 4 (c) 3 (d) 5
18. 18.What is the smallest number you must add to 101 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
19. 19.What is the units digit of 3^54? (a) 1 (b) 7 (c) 9 (d) 3
20. 20.What is the smallest number you must add to 747 to make it divisible by 12? (a) 9 (b) 8 (c) 3 (d) 10
21. 21.What is the units digit of 7^34? (a) 3 (b) 1 (c) 7 (d) 9
22. 22.What is the units digit of 2^18? (a) 8 (b) 4 (c) 6 (d) 2
23. 23.What is the remainder when 321 is divided by 12? (a) 8 (b) 10 (c) 3 (d) 9
24. 24.What is the remainder when 70 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
25. 25.Which of these numbers is divisible by 4? (a) 2703 (b) 2697 (c) 2696 (d) 2693
26. 26.What is the remainder when 259 is divided by 8? (a) 4 (b) 5 (c) 2 (d) 3
27. 27.What is the remainder when 488 is divided by 7? (a) 5 (b) 4 (c) 6 (d) 2
28. 28.What is the remainder when 221 is divided by 5? (a) 0 (b) 2 (c) 1 (d) 4
29. 29.What is the smallest number you must add to 647 to make it divisible by 12? (a) 11 (b) 3 (c) 2 (d) 1
30. 30.What is the smallest number you must add to 254 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 2
31. 31.What is the remainder when 394 is divided by 12? (a) 2 (b) 10 (c) 11 (d) 9
32. 32.Which of these numbers is divisible by 9? (a) 5920 (b) 5923 (c) 5922 (d) 5939
33. 33.What is the units digit of 2^52? (a) 4 (b) 6 (c) 2 (d) 8
34. 34.Which of these numbers is divisible by 8? (a) 5008 (b) 5017 (c) 5003 (d) 5015
35. 35.Which of these numbers is divisible by 3? (a) 590 (b) 588 (c) 586 (d) 592
36. 36.What is the smallest number you must add to 218 to make it divisible by 6? (a) 5 (b) 2 (c) 4 (d) 3
37. 37.What is the remainder when 917 is divided by 12? (a) 7 (b) 5 (c) 6 (d) 4
38. 38.Which of these numbers is divisible by 8? (a) 3479 (b) 3466 (c) 3459 (d) 3464
39. 39.Which of these numbers is divisible by 6? (a) 2983 (b) 2989 (c) 2982 (d) 2980
40. 40.What is the units digit of 4^57? (a) 3 (b) 5 (c) 6 (d) 4

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