Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 99 · 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 units digit of 7^33? (a) 1 (b) 3 (c) 7 (d) 9
2. 2.What is the remainder when 337 is divided by 8? (a) 7 (b) 1 (c) 0 (d) 2
3. 3.What is the remainder when 146 is divided by 9? (a) 1 (b) 3 (c) 7 (d) 2
4. 4.Which of these numbers is divisible by 6? (a) 2555 (b) 2540 (c) 2546 (d) 2544
5. 5.What is the units digit of 7^28? (a) 1 (b) 3 (c) 7 (d) 9
6. 6.What is the smallest number you must add to 96 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
7. 7.What is the remainder when 88 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
8. 8.What is the units digit of 3^15? (a) 7 (b) 1 (c) 9 (d) 3
9. 9.Which of these numbers is divisible by 11? (a) 6042 (b) 6022 (c) 6028 (d) 6031
10. 10.Which of these numbers is divisible by 3? (a) 1418 (b) 1420 (c) 1419 (d) 1424
11. 11.Which of these numbers is divisible by 11? (a) 6117 (b) 6128 (c) 6127 (d) 6147
12. 12.What is the units digit of 7^23? (a) 7 (b) 9 (c) 1 (d) 3
13. 13.What is the units digit of 7^25? (a) 1 (b) 3 (c) 7 (d) 9
14. 14.What is the smallest number you must add to 184 to make it divisible by 3? (a) 2 (b) 1
15. 15.What is the remainder when 251 is divided by 11? (a) 10 (b) 9 (c) 8 (d) 2
16. 16.What is the smallest number you must add to 539 to make it divisible by 8? (a) 6 (b) 4 (c) 3 (d) 5
17. 17.What is the remainder when 386 is divided by 5? (a) 0 (b) 4 (c) 1 (d) 2
18. 18.Which of these numbers is divisible by 6? (a) 2632 (b) 2627 (c) 2622 (d) 2619
19. 19.What is the remainder when 380 is divided by 12? (a) 4 (b) 9 (c) 8 (d) 7
20. 20.What is the smallest number you must add to 399 to make it divisible by 10? (a) 2 (b) 3 (c) 9 (d) 1
21. 21.What is the smallest number you must add to 197 to make it divisible by 8? (a) 2 (b) 5 (c) 3 (d) 4
22. 22.What is the remainder when 381 is divided by 6? (a) 3 (b) 5 (c) 2 (d) 4
23. 23.Which of these numbers is divisible by 4? (a) 3214 (b) 3208 (c) 3211 (d) 3206
24. 24.What is the smallest number you must add to 331 to make it divisible by 10? (a) 1 (b) 7 (c) 9 (d) 8
25. 25.What is the units digit of 7^38? (a) 1 (b) 7 (c) 3 (d) 9
26. 26.What is the smallest number you must add to 477 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 2
27. 27.Which of these numbers is divisible by 9? (a) 394 (b) 377 (c) 378 (d) 380
28. 28.What is the smallest number you must add to 166 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
29. 29.Which of these numbers is divisible by 11? (a) 3133 (b) 3142 (c) 3115 (d) 3124
30. 30.What is the smallest number you must add to 342 to make it divisible by 11? (a) 1 (b) 10 (c) 8 (d) 9
31. 31.What is the remainder when 329 is divided by 9? (a) 7 (b) 6 (c) 4 (d) 5
32. 32.What is the smallest number you must add to 194 to make it divisible by 8? (a) 7 (b) 2 (c) 6 (d) 5
33. 33.Which of these numbers is divisible by 6? (a) 3665 (b) 3658 (c) 3671 (d) 3660
34. 34.Which of these numbers is divisible by 4? (a) 3277 (b) 3276 (c) 3281 (d) 3274
35. 35.What is the units digit of 7^42? (a) 3 (b) 9 (c) 1 (d) 7
36. 36.What is the units digit of 3^17? (a) 3 (b) 1 (c) 9 (d) 7
37. 37.What is the units digit of 7^56? (a) 7 (b) 9 (c) 3 (d) 1
38. 38.What is the units digit of 9^18? (a) 1 (b) 9 (c) 2 (d) 0
39. 39.What is the remainder when 511 is divided by 12? (a) 5 (b) 7 (c) 6 (d) 8
40. 40.What is the remainder when 160 is divided by 11? (a) 7 (b) 8 (c) 5 (d) 6

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