Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 616 · 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 288 is divided by 7? (a) 6 (b) 2 (c) 1 (d) 0
2. 2.Which of these numbers is divisible by 3? (a) 1991 (b) 1993 (c) 1989 (d) 1988
3. 3.What is the smallest number you must add to 134 to make it divisible by 4? (a) 1 (b) 2 (c) 3
4. 4.Which of these numbers is divisible by 4? (a) 3087 (b) 3082 (c) 3091 (d) 3084
5. 5.What is the remainder when 376 is divided by 5? (a) 1 (b) 4 (c) 0 (d) 2
6. 6.What is the remainder when 322 is divided by 8? (a) 3 (b) 1 (c) 6 (d) 2
7. 7.What is the smallest number you must add to 217 to make it divisible by 8? (a) 7 (b) 6 (c) 5 (d) 1
8. 8.What is the units digit of 9^30? (a) 1 (b) 0 (c) 2 (d) 9
9. 9.What is the units digit of 8^47? (a) 6 (b) 8 (c) 4 (d) 2
10. 10.Which of these numbers is divisible by 8? (a) 6432 (b) 6426 (c) 6436 (d) 6442
11. 11.Which of these numbers is divisible by 6? (a) 5331 (b) 5335 (c) 5343 (d) 5334
12. 12.Which of these numbers is divisible by 8? (a) 855 (b) 842 (c) 862 (d) 848
13. 13.Which of these numbers is divisible by 11? (a) 2233 (b) 2228 (c) 2240 (d) 2246
14. 14.What is the smallest number you must add to 271 to make it divisible by 10? (a) 8 (b) 7 (c) 9 (d) 1
15. 15.Which of these numbers is divisible by 3? (a) 2420 (b) 2422 (c) 2421 (d) 2425
16. 16.What is the remainder when 722 is divided by 11? (a) 4 (b) 8 (c) 7 (d) 6
17. 17.Which of these numbers is divisible by 3? (a) 2600 (b) 2601 (c) 2605 (d) 2603
18. 18.What is the smallest number you must add to 367 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 5
19. 19.What is the remainder when 52 is divided by 3? (a) 0 (b) 2 (c) 1
20. 20.What is the remainder when 106 is divided by 9? (a) 2 (b) 7 (c) 8 (d) 6
21. 21.What is the units digit of 8^39? (a) 2 (b) 4 (c) 6 (d) 8
22. 22.Which of these numbers is divisible by 3? (a) 1666 (b) 1669 (c) 1672 (d) 1668
23. 23.What is the smallest number you must add to 690 to make it divisible by 9? (a) 3 (b) 2 (c) 6 (d) 4
24. 24.What is the units digit of 2^25? (a) 4 (b) 2 (c) 8 (d) 6
25. 25.What is the smallest number you must add to 306 to make it divisible by 4? (a) 1 (b) 3 (c) 2
26. 26.What is the remainder when 180 is divided by 8? (a) 3 (b) 5 (c) 4 (d) 6
27. 27.What is the remainder when 651 is divided by 9? (a) 3 (b) 4 (c) 2 (d) 6
28. 28.What is the units digit of 2^12? (a) 4 (b) 2 (c) 6 (d) 8
29. 29.What is the units digit of 3^14? (a) 7 (b) 3 (c) 9 (d) 1
30. 30.Which of these numbers is divisible by 9? (a) 3330 (b) 3342 (c) 3328 (d) 3333
31. 31.What is the smallest number you must add to 381 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
32. 32.What is the units digit of 2^45? (a) 4 (b) 6 (c) 2 (d) 8
33. 33.What is the remainder when 222 is divided by 9? (a) 6 (b) 5 (c) 3 (d) 7
34. 34.What is the units digit of 4^30? (a) 4 (b) 6 (c) 5 (d) 7
35. 35.What is the units digit of 2^38? (a) 2 (b) 8 (c) 4 (d) 6
36. 36.What is the smallest number you must add to 385 to make it divisible by 9? (a) 3 (b) 2 (c) 7 (d) 1
37. 37.What is the remainder when 88 is divided by 3? (a) 2 (b) 0 (c) 1
38. 38.What is the units digit of 3^22? (a) 3 (b) 7 (c) 1 (d) 9
39. 39.What is the units digit of 8^10? (a) 8 (b) 6 (c) 4 (d) 2
40. 40.What is the remainder when 131 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3

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