Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 283 · 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 169 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
2. 2.What is the smallest number you must add to 203 to make it divisible by 12? (a) 3 (b) 11 (c) 1 (d) 2
3. 3.What is the smallest number you must add to 182 to make it divisible by 10? (a) 8 (b) 2 (c) 9 (d) 7
4. 4.What is the smallest number you must add to 173 to make it divisible by 4? (a) 2 (b) 3 (c) 1
5. 5.What is the smallest number you must add to 133 to make it divisible by 4? (a) 3 (b) 1 (c) 2
6. 6.What is the units digit of 3^54? (a) 3 (b) 1 (c) 9 (d) 7
7. 7.What is the remainder when 319 is divided by 6? (a) 2 (b) 0 (c) 1 (d) 5
8. 8.What is the smallest number you must add to 493 to make it divisible by 9? (a) 7 (b) 3 (c) 1 (d) 2
9. 9.What is the remainder when 66 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
10. 10.What is the units digit of 9^32? (a) 2 (b) 1 (c) 9 (d) 0
11. 11.What is the smallest number you must add to 43 to make it divisible by 3? (a) 2 (b) 1
12. 12.Which of these numbers is divisible by 4? (a) 1327 (b) 1329 (c) 1328 (d) 1334
13. 13.What is the remainder when 104 is divided by 10? (a) 6 (b) 5 (c) 3 (d) 4
14. 14.What is the remainder when 519 is divided by 7? (a) 0 (b) 2 (c) 6 (d) 1
15. 15.Which of these numbers is divisible by 11? (a) 2843 (b) 2838 (c) 2851 (d) 2832
16. 16.Which of these numbers is divisible by 6? (a) 5034 (b) 5029 (c) 5045 (d) 5039
17. 17.Which of these numbers is divisible by 8? (a) 2881 (b) 2888 (c) 2901 (d) 2894
18. 18.What is the remainder when 158 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
19. 19.Which of these numbers is divisible by 3? (a) 548 (b) 545 (c) 546 (d) 551
20. 20.Which of these numbers is divisible by 9? (a) 3420 (b) 3414 (c) 3425 (d) 3435
21. 21.What is the smallest number you must add to 257 to make it divisible by 7? (a) 3 (b) 1 (c) 5 (d) 2
22. 22.What is the smallest number you must add to 265 to make it divisible by 8? (a) 7 (b) 1 (c) 5 (d) 6
23. 23.Which of these numbers is divisible by 11? (a) 982 (b) 1008 (c) 990 (d) 998
24. 24.What is the units digit of 2^22? (a) 6 (b) 2 (c) 8 (d) 4
25. 25.Which of these numbers is divisible by 9? (a) 2403 (b) 2408 (c) 2396 (d) 2413
26. 26.What is the smallest number you must add to 113 to make it divisible by 8? (a) 6 (b) 5 (c) 1 (d) 7
27. 27.Which of these numbers is divisible by 9? (a) 4925 (b) 4938 (c) 4923 (d) 4916
28. 28.What is the units digit of 7^54? (a) 1 (b) 9 (c) 7 (d) 3
29. 29.What is the units digit of 7^57? (a) 7 (b) 1 (c) 9 (d) 3
30. 30.Which of these numbers is divisible by 3? (a) 677 (b) 670 (c) 672 (d) 673
31. 31.Which of these numbers is divisible by 6? (a) 1036 (b) 1041 (c) 1032 (d) 1028
32. 32.Which of these numbers is divisible by 8? (a) 2188 (b) 2183 (c) 2199 (d) 2184
33. 33.What is the units digit of 7^40? (a) 3 (b) 1 (c) 9 (d) 7
34. 34.What is the units digit of 9^47? (a) 8 (b) 7 (c) 9 (d) 1
35. 35.What is the units digit of 4^49? (a) 4 (b) 5 (c) 3 (d) 6
36. 36.What is the units digit of 9^38? (a) 2 (b) 9 (c) 1 (d) 0
37. 37.What is the units digit of 7^38? (a) 7 (b) 1 (c) 3 (d) 9
38. 38.Which of these numbers is divisible by 3? (a) 2288 (b) 2290 (c) 2294 (d) 2289
39. 39.Which of these numbers is divisible by 9? (a) 1611 (b) 1603 (c) 1614 (d) 1623
40. 40.What is the remainder when 617 is divided by 8? (a) 7 (b) 0 (c) 2 (d) 1

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