Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 119 · 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 451 is divided by 12? (a) 7 (b) 6 (c) 5 (d) 8
2. 2.Which of these numbers is divisible by 8? (a) 3762 (b) 3754 (c) 3760 (d) 3769
3. 3.What is the units digit of 2^30? (a) 8 (b) 2 (c) 4 (d) 6
4. 4.What is the smallest number you must add to 78 to make it divisible by 4? (a) 1 (b) 3 (c) 2
5. 5.What is the smallest number you must add to 493 to make it divisible by 10? (a) 7 (b) 8 (c) 3 (d) 6
6. 6.What is the smallest number you must add to 459 to make it divisible by 8? (a) 6 (b) 4 (c) 3 (d) 5
7. 7.What is the remainder when 82 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
8. 8.What is the units digit of 7^53? (a) 9 (b) 3 (c) 1 (d) 7
9. 9.What is the smallest number you must add to 247 to make it divisible by 6? (a) 5 (b) 3 (c) 4 (d) 1
10. 10.Which of these numbers is divisible by 3? (a) 1053 (b) 1054 (c) 1051 (d) 1058
11. 11.Which of these numbers is divisible by 3? (a) 2291 (b) 2293 (c) 2287 (d) 2289
12. 12.What is the units digit of 4^18? (a) 6 (b) 4 (c) 5 (d) 7
13. 13.What is the smallest number you must add to 248 to make it divisible by 10? (a) 3 (b) 1 (c) 8 (d) 2
14. 14.What is the remainder when 138 is divided by 10? (a) 9 (b) 8 (c) 7 (d) 2
15. 15.What is the units digit of 8^28? (a) 4 (b) 8 (c) 2 (d) 6
16. 16.What is the units digit of 4^48? (a) 7 (b) 5 (c) 4 (d) 6
17. 17.What is the units digit of 9^51? (a) 7 (b) 8 (c) 1 (d) 9
18. 18.What is the remainder when 95 is divided by 9? (a) 5 (b) 6 (c) 4 (d) 7
19. 19.What is the remainder when 287 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
20. 20.What is the smallest number you must add to 259 to make it divisible by 4? (a) 2 (b) 1 (c) 3
21. 21.What is the units digit of 2^12? (a) 8 (b) 6 (c) 2 (d) 4
22. 22.Which of these numbers is divisible by 8? (a) 4811 (b) 4820 (c) 4808 (d) 4801
23. 23.Which of these numbers is divisible by 6? (a) 320 (b) 312 (c) 310 (d) 315
24. 24.What is the remainder when 366 is divided by 11? (a) 4 (b) 2 (c) 8 (d) 3
25. 25.What is the units digit of 3^17? (a) 9 (b) 1 (c) 7 (d) 3
26. 26.Which of these numbers is divisible by 3? (a) 2137 (b) 2140 (c) 2144 (d) 2139
27. 27.Which of these numbers is divisible by 11? (a) 5200 (b) 5213 (c) 5192 (d) 5190
28. 28.What is the units digit of 9^28? (a) 9 (b) 1 (c) 0 (d) 2
29. 29.Which of these numbers is divisible by 3? (a) 412 (b) 416 (c) 409 (d) 411
30. 30.What is the smallest number you must add to 322 to make it divisible by 6? (a) 3 (b) 4 (c) 2 (d) 1
31. 31.What is the smallest number you must add to 215 to make it divisible by 4? (a) 3 (b) 2 (c) 1
32. 32.Which of these numbers is divisible by 8? (a) 1327 (b) 1314 (c) 1320 (d) 1330
33. 33.Which of these numbers is divisible by 8? (a) 1234 (b) 1242 (c) 1230 (d) 1232
34. 34.Which of these numbers is divisible by 8? (a) 404 (b) 392 (c) 397 (d) 386
35. 35.Which of these numbers is divisible by 11? (a) 1675 (b) 1683 (c) 1692 (d) 1704
36. 36.What is the remainder when 251 is divided by 5? (a) 0 (b) 4 (c) 1 (d) 2
37. 37.What is the smallest number you must add to 73 to make it divisible by 4? (a) 3 (b) 1 (c) 2
38. 38.What is the remainder when 163 is divided by 7? (a) 1 (b) 2 (c) 5 (d) 3
39. 39.Which of these numbers is divisible by 9? (a) 3870 (b) 3880 (c) 3871 (d) 3863
40. 40.Which of these numbers is divisible by 3? (a) 2422 (b) 2418 (c) 2419 (d) 2417

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