Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 664 · 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) 3823 (b) 3832 (c) 3820 (d) 3822
2. 2.What is the remainder when 596 is divided by 12? (a) 8 (b) 7 (c) 9 (d) 4
3. 3.What is the smallest number you must add to 360 to make it divisible by 7? (a) 3 (b) 4 (c) 6 (d) 5
4. 4.What is the units digit of 4^50? (a) 4 (b) 7 (c) 5 (d) 6
5. 5.Which of these numbers is divisible by 9? (a) 4472 (b) 4464 (c) 4476 (d) 4462
6. 6.What is the smallest number you must add to 128 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
7. 7.Which of these numbers is divisible by 3? (a) 2555 (b) 2553 (c) 2558 (d) 2551
8. 8.What is the units digit of 8^14? (a) 4 (b) 8 (c) 2 (d) 6
9. 9.Which of these numbers is divisible by 3? (a) 1774 (b) 1777 (c) 1781 (d) 1776
10. 10.Which of these numbers is divisible by 11? (a) 6124 (b) 6116 (c) 6115 (d) 6132
11. 11.What is the units digit of 9^58? (a) 9 (b) 1 (c) 0 (d) 2
12. 12.What is the smallest number you must add to 232 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
13. 13.Which of these numbers is divisible by 3? (a) 1637 (b) 1632 (c) 1634 (d) 1631
14. 14.What is the smallest number you must add to 389 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
15. 15.What is the remainder when 168 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
16. 16.What is the remainder when 113 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
17. 17.What is the units digit of 7^12? (a) 7 (b) 9 (c) 1 (d) 3
18. 18.Which of these numbers is divisible by 4? (a) 442 (b) 445 (c) 450 (d) 444
19. 19.What is the units digit of 9^37? (a) 7 (b) 1 (c) 8 (d) 9
20. 20.What is the smallest number you must add to 255 to make it divisible by 6? (a) 3 (b) 2 (c) 5 (d) 4
21. 21.What is the smallest number you must add to 650 to make it divisible by 9? (a) 6 (b) 8 (c) 2 (d) 7
22. 22.What is the smallest number you must add to 251 to make it divisible by 7? (a) 6 (b) 1 (c) 2 (d) 3
23. 23.What is the remainder when 191 is divided by 7? (a) 5 (b) 2 (c) 3 (d) 1
24. 24.Which of these numbers is divisible by 4? (a) 2583 (b) 2576 (c) 2573 (d) 2579
25. 25.Which of these numbers is divisible by 4? (a) 423 (b) 429 (c) 427 (d) 424
26. 26.What is the remainder when 229 is divided by 3? (a) 2 (b) 1 (c) 0
27. 27.What is the remainder when 328 is divided by 7? (a) 4 (b) 1 (c) 5 (d) 6
28. 28.What is the units digit of 9^20? (a) 9 (b) 2 (c) 1 (d) 0
29. 29.What is the units digit of 8^10? (a) 8 (b) 2 (c) 4 (d) 6
30. 30.Which of these numbers is divisible by 8? (a) 2144 (b) 2147 (c) 2140 (d) 2159
31. 31.What is the remainder when 138 is divided by 10? (a) 9 (b) 8 (c) 2 (d) 7
32. 32.What is the remainder when 243 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
33. 33.Which of these numbers is divisible by 4? (a) 1318 (b) 1315 (c) 1323 (d) 1316
34. 34.Which of these numbers is divisible by 9? (a) 5901 (b) 5890 (c) 5895 (d) 5905
35. 35.What is the remainder when 664 is divided by 12? (a) 5 (b) 3 (c) 4 (d) 8
36. 36.What is the smallest number you must add to 353 to make it divisible by 7? (a) 6 (b) 3 (c) 5 (d) 4
37. 37.Which of these numbers is divisible by 8? (a) 6961 (b) 6952 (c) 6948 (d) 6955
38. 38.What is the smallest number you must add to 265 to make it divisible by 4? (a) 2 (b) 1 (c) 3
39. 39.What is the remainder when 43 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
40. 40.Which of these numbers is divisible by 3? (a) 959 (b) 960 (c) 964 (d) 962

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