Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 916 · 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 smallest number you must add to 340 to make it divisible by 12? (a) 9 (b) 4 (c) 8 (d) 7
2. 2.What is the remainder when 267 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
3. 3.What is the smallest number you must add to 378 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
4. 4.Which of these numbers is divisible by 3? (a) 1268 (b) 1274 (c) 1270 (d) 1269
5. 5.Which of these numbers is divisible by 6? (a) 4235 (b) 4224 (c) 4227 (d) 4221
6. 6.Which of these numbers is divisible by 11? (a) 8717 (b) 8712 (c) 8729 (d) 8702
7. 7.What is the units digit of 9^22? (a) 1 (b) 9 (c) 2 (d) 0
8. 8.What is the smallest number you must add to 305 to make it divisible by 8? (a) 6 (b) 1 (c) 7 (d) 5
9. 9.What is the units digit of 3^46? (a) 7 (b) 3 (c) 9 (d) 1
10. 10.What is the remainder when 363 is divided by 6? (a) 4 (b) 2 (c) 5 (d) 3
11. 11.What is the smallest number you must add to 181 to make it divisible by 6? (a) 3 (b) 4 (c) 1 (d) 5
12. 12.Which of these numbers is divisible by 4? (a) 3446 (b) 3448 (c) 3453 (d) 3451
13. 13.What is the remainder when 217 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
14. 14.What is the smallest number you must add to 137 to make it divisible by 4? (a) 3 (b) 2 (c) 1
15. 15.What is the units digit of 8^39? (a) 8 (b) 6 (c) 2 (d) 4
16. 16.What is the units digit of 8^44? (a) 4 (b) 2 (c) 8 (d) 6
17. 17.What is the remainder when 281 is divided by 5? (a) 4 (b) 2 (c) 0 (d) 1
18. 18.What is the smallest number you must add to 97 to make it divisible by 8? (a) 5 (b) 7 (c) 1 (d) 6
19. 19.What is the smallest number you must add to 785 to make it divisible by 12? (a) 5 (b) 7 (c) 8 (d) 6
20. 20.What is the remainder when 61 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
21. 21.Which of these numbers is divisible by 9? (a) 6747 (b) 6760 (c) 6752 (d) 6750
22. 22.What is the smallest number you must add to 37 to make it divisible by 3? (a) 2 (b) 1
23. 23.What is the remainder when 245 is divided by 8? (a) 4 (b) 3 (c) 5 (d) 6
24. 24.Which of these numbers is divisible by 9? (a) 6810 (b) 6789 (c) 6795 (d) 6802
25. 25.What is the smallest number you must add to 273 to make it divisible by 12? (a) 2 (b) 9 (c) 3 (d) 4
26. 26.What is the units digit of 2^52? (a) 8 (b) 4 (c) 6 (d) 2
27. 27.Which of these numbers is divisible by 8? (a) 3896 (b) 3899 (c) 3894 (d) 3905
28. 28.Which of these numbers is divisible by 11? (a) 1951 (b) 1964 (c) 1943 (d) 1947
29. 29.What is the units digit of 2^43? (a) 8 (b) 4 (c) 6 (d) 2
30. 30.Which of these numbers is divisible by 6? (a) 4292 (b) 4290 (c) 4300 (d) 4287
31. 31.What is the remainder when 124 is divided by 3? (a) 0 (b) 1 (c) 2
32. 32.What is the smallest number you must add to 93 to make it divisible by 7? (a) 4 (b) 6 (c) 5 (d) 2
33. 33.What is the smallest number you must add to 204 to make it divisible by 10? (a) 7 (b) 6 (c) 4 (d) 5
34. 34.What is the units digit of 3^10? (a) 1 (b) 7 (c) 3 (d) 9
35. 35.What is the remainder when 513 is divided by 11? (a) 8 (b) 6 (c) 7 (d) 4
36. 36.What is the units digit of 4^10? (a) 4 (b) 6 (c) 5 (d) 7
37. 37.What is the remainder when 590 is divided by 11? (a) 4 (b) 6 (c) 7 (d) 8
38. 38.What is the smallest number you must add to 857 to make it divisible by 12? (a) 8 (b) 5 (c) 6 (d) 7
39. 39.What is the units digit of 2^27? (a) 4 (b) 6 (c) 8 (d) 2
40. 40.Which of these numbers is divisible by 11? (a) 1991 (b) 1995 (c) 1989 (d) 2003

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