Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 399 · 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 194 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
2. 2.What is the units digit of 7^38? (a) 9 (b) 7 (c) 3 (d) 1
3. 3.Which of these numbers is divisible by 9? (a) 6241 (b) 6251 (c) 6260 (d) 6246
4. 4.What is the units digit of 8^33? (a) 8 (b) 6 (c) 4 (d) 2
5. 5.What is the smallest number you must add to 267 to make it divisible by 8? (a) 3 (b) 4 (c) 6 (d) 5
6. 6.What is the remainder when 148 is divided by 3? (a) 2 (b) 0 (c) 1
7. 7.What is the smallest number you must add to 72 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
8. 8.What is the remainder when 339 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
9. 9.What is the remainder when 271 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
10. 10.Which of these numbers is divisible by 11? (a) 9577 (b) 9562 (c) 9553 (d) 9559
11. 11.What is the remainder when 178 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
12. 12.Which of these numbers is divisible by 6? (a) 1236 (b) 1247 (c) 1235 (d) 1241
13. 13.What is the units digit of 2^30? (a) 2 (b) 6 (c) 8 (d) 4
14. 14.What is the smallest number you must add to 394 to make it divisible by 9? (a) 2 (b) 1 (c) 7 (d) 3
15. 15.What is the smallest number you must add to 365 to make it divisible by 7? (a) 6 (b) 5 (c) 1 (d) 4
16. 16.What is the smallest number you must add to 283 to make it divisible by 6? (a) 1 (b) 3 (c) 4 (d) 5
17. 17.What is the units digit of 2^38? (a) 6 (b) 2 (c) 4 (d) 8
18. 18.Which of these numbers is divisible by 11? (a) 1835 (b) 1837 (c) 1840 (d) 1849
19. 19.What is the smallest number you must add to 153 to make it divisible by 4? (a) 2 (b) 1 (c) 3
20. 20.What is the remainder when 352 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
21. 21.What is the smallest number you must add to 284 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
22. 22.What is the units digit of 7^35? (a) 7 (b) 9 (c) 1 (d) 3
23. 23.What is the remainder when 357 is divided by 6? (a) 3 (b) 5 (c) 4 (d) 2
24. 24.What is the units digit of 9^52? (a) 9 (b) 1 (c) 0 (d) 2
25. 25.Which of these numbers is divisible by 3? (a) 1472 (b) 1474 (c) 1470 (d) 1468
26. 26.What is the remainder when 317 is divided by 8? (a) 3 (b) 4 (c) 6 (d) 5
27. 27.What is the units digit of 9^25? (a) 1 (b) 9 (c) 8 (d) 7
28. 28.What is the remainder when 706 is divided by 11? (a) 3 (b) 1 (c) 9 (d) 2
29. 29.What is the units digit of 4^39? (a) 3 (b) 6 (c) 4 (d) 5
30. 30.What is the smallest number you must add to 163 to make it divisible by 11? (a) 3 (b) 9 (c) 2 (d) 1
31. 31.What is the units digit of 3^28? (a) 3 (b) 9 (c) 1 (d) 7
32. 32.What is the units digit of 7^19? (a) 9 (b) 3 (c) 1 (d) 7
33. 33.What is the smallest number you must add to 251 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
34. 34.What is the smallest number you must add to 182 to make it divisible by 9? (a) 8 (b) 2 (c) 7 (d) 6
35. 35.What is the remainder when 656 is divided by 11? (a) 4 (b) 7 (c) 6 (d) 8
36. 36.What is the smallest number you must add to 183 to make it divisible by 10? (a) 7 (b) 6 (c) 3 (d) 8
37. 37.What is the units digit of 7^14? (a) 3 (b) 7 (c) 9 (d) 1
38. 38.Which of these numbers is divisible by 4? (a) 267 (b) 260 (c) 258 (d) 263
39. 39.What is the smallest number you must add to 374 to make it divisible by 12? (a) 9 (b) 2 (c) 11 (d) 10
40. 40.Which of these numbers is divisible by 6? (a) 2724 (b) 2729 (c) 2719 (d) 2733

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