Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 724 · 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 3? (a) 1885 (b) 1881 (c) 1883 (d) 1880
2. 2.What is the smallest number you must add to 186 to make it divisible by 10? (a) 5 (b) 4 (c) 3 (d) 6
3. 3.Which of these numbers is divisible by 4? (a) 3379 (b) 3380 (c) 3386 (d) 3382
4. 4.What is the remainder when 252 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
5. 5.Which of these numbers is divisible by 4? (a) 1736 (b) 1733 (c) 1738 (d) 1743
6. 6.Which of these numbers is divisible by 9? (a) 2296 (b) 2307 (c) 2321 (d) 2304
7. 7.What is the units digit of 9^39? (a) 9 (b) 7 (c) 8 (d) 1
8. 8.Which of these numbers is divisible by 8? (a) 6815 (b) 6816 (c) 6817 (d) 6828
9. 9.What is the smallest number you must add to 310 to make it divisible by 8? (a) 3 (b) 6 (c) 1 (d) 2
10. 10.What is the units digit of 8^54? (a) 6 (b) 4 (c) 2 (d) 8
11. 11.What is the remainder when 159 is divided by 8? (a) 7 (b) 1 (c) 5 (d) 6
12. 12.What is the remainder when 367 is divided by 8? (a) 1 (b) 6 (c) 7 (d) 5
13. 13.What is the units digit of 4^20? (a) 6 (b) 5 (c) 7 (d) 4
14. 14.What is the smallest number you must add to 265 to make it divisible by 4? (a) 1 (b) 3 (c) 2
15. 15.What is the remainder when 554 is divided by 8? (a) 2 (b) 1 (c) 6 (d) 3
16. 16.Which of these numbers is divisible by 3? (a) 1600 (b) 1595 (c) 1598 (d) 1596
17. 17.What is the smallest number you must add to 299 to make it divisible by 10? (a) 9 (b) 3 (c) 2 (d) 1
18. 18.What is the units digit of 2^21? (a) 4 (b) 8 (c) 6 (d) 2
19. 19.Which of these numbers is divisible by 6? (a) 3371 (b) 3383 (c) 3372 (d) 3374
20. 20.What is the smallest number you must add to 206 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
21. 21.What is the units digit of 4^31? (a) 5 (b) 6 (c) 3 (d) 4
22. 22.Which of these numbers is divisible by 9? (a) 6895 (b) 6894 (c) 6905 (d) 6887
23. 23.What is the remainder when 267 is divided by 9? (a) 3 (b) 6 (c) 7 (d) 5
24. 24.What is the remainder when 336 is divided by 5? (a) 1 (b) 2 (c) 0 (d) 4
25. 25.What is the smallest number you must add to 455 to make it divisible by 8? (a) 2 (b) 3 (c) 1 (d) 7
26. 26.What is the remainder when 160 is divided by 6? (a) 3 (b) 5 (c) 4 (d) 2
27. 27.What is the remainder when 617 is divided by 8? (a) 0 (b) 2 (c) 7 (d) 1
28. 28.Which of these numbers is divisible by 8? (a) 6344 (b) 6346 (c) 6357 (d) 6339
29. 29.What is the remainder when 235 is divided by 6? (a) 0 (b) 1 (c) 2 (d) 5
30. 30.What is the units digit of 9^31? (a) 8 (b) 9 (c) 1 (d) 7
31. 31.What is the smallest number you must add to 104 to make it divisible by 3? (a) 2 (b) 1
32. 32.What is the units digit of 8^58? (a) 6 (b) 4 (c) 2 (d) 8
33. 33.What is the smallest number you must add to 269 to make it divisible by 10? (a) 9 (b) 2 (c) 3 (d) 1
34. 34.Which of these numbers is divisible by 4? (a) 3289 (b) 3298 (c) 3293 (d) 3292
35. 35.Which of these numbers is divisible by 11? (a) 9890 (b) 9878 (c) 9888 (d) 9873
36. 36.What is the remainder when 468 is divided by 7? (a) 5 (b) 6 (c) 1 (d) 4
37. 37.What is the smallest number you must add to 92 to make it divisible by 3? (a) 2 (b) 1
38. 38.Which of these numbers is divisible by 4? (a) 198 (b) 195 (c) 201 (d) 196
39. 39.What is the remainder when 158 is divided by 3? (a) 0 (b) 2 (c) 1
40. 40.What is the remainder when 227 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0

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