Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 100 · 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 2^58? (a) 2 (b) 8 (c) 6 (d) 4
2. 2.What is the smallest number you must add to 199 to make it divisible by 3? (a) 2 (b) 1
3. 3.What is the remainder when 238 is divided by 8? (a) 7 (b) 2 (c) 5 (d) 6
4. 4.Which of these numbers is divisible by 3? (a) 2626 (b) 2630 (c) 2623 (d) 2625
5. 5.What is the units digit of 2^29? (a) 8 (b) 6 (c) 4 (d) 2
6. 6.What is the remainder when 170 is divided by 6? (a) 4 (b) 2 (c) 1 (d) 3
7. 7.What is the smallest number you must add to 243 to make it divisible by 10? (a) 3 (b) 8 (c) 7 (d) 6
8. 8.What is the units digit of 3^31? (a) 9 (b) 1 (c) 7 (d) 3
9. 9.What is the remainder when 213 is divided by 12? (a) 3 (b) 9 (c) 10 (d) 8
10. 10.What is the remainder when 724 is divided by 10? (a) 5 (b) 6 (c) 3 (d) 4
11. 11.What is the remainder when 91 is divided by 5? (a) 0 (b) 4 (c) 2 (d) 1
12. 12.Which of these numbers is divisible by 4? (a) 546 (b) 544 (c) 549 (d) 541
13. 13.What is the smallest number you must add to 263 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
14. 14.Which of these numbers is divisible by 4? (a) 1093 (b) 1096 (c) 1102 (d) 1099
15. 15.What is the units digit of 3^23? (a) 3 (b) 7 (c) 1 (d) 9
16. 16.What is the units digit of 2^42? (a) 4 (b) 8 (c) 2 (d) 6
17. 17.Which of these numbers is divisible by 6? (a) 4023 (b) 4014 (c) 4009 (d) 4015
18. 18.Which of these numbers is divisible by 3? (a) 2611 (b) 2614 (c) 2618 (d) 2613
19. 19.What is the units digit of 3^18? (a) 9 (b) 1 (c) 3 (d) 7
20. 20.What is the remainder when 169 is divided by 3? (a) 0 (b) 2 (c) 1
21. 21.What is the remainder when 371 is divided by 8? (a) 3 (b) 4 (c) 2 (d) 5
22. 22.What is the smallest number you must add to 331 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
23. 23.What is the smallest number you must add to 142 to make it divisible by 4? (a) 2 (b) 1 (c) 3
24. 24.Which of these numbers is divisible by 3? (a) 1252 (b) 1254 (c) 1256 (d) 1258
25. 25.What is the remainder when 177 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
26. 26.What is the smallest number you must add to 344 to make it divisible by 7? (a) 5 (b) 4 (c) 1 (d) 6
27. 27.What is the smallest number you must add to 131 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
28. 28.Which of these numbers is divisible by 6? (a) 1164 (b) 1165 (c) 1163 (d) 1172
29. 29.Which of these numbers is divisible by 3? (a) 966 (b) 964 (c) 971 (d) 967
30. 30.What is the remainder when 339 is divided by 7? (a) 2 (b) 4 (c) 3 (d) 5
31. 31.Which of these numbers is divisible by 9? (a) 5503 (b) 5489 (c) 5490 (d) 5493
32. 32.Which of these numbers is divisible by 9? (a) 4572 (b) 4587 (c) 4566 (d) 4580
33. 33.Which of these numbers is divisible by 4? (a) 1826 (b) 1818 (c) 1822 (d) 1820
34. 34.What is the units digit of 3^35? (a) 9 (b) 1 (c) 3 (d) 7
35. 35.Which of these numbers is divisible by 4? (a) 694 (b) 685 (c) 691 (d) 688
36. 36.What is the units digit of 2^47? (a) 6 (b) 4 (c) 2 (d) 8
37. 37.Which of these numbers is divisible by 6? (a) 3410 (b) 3402 (c) 3398 (d) 3404
38. 38.Which of these numbers is divisible by 4? (a) 3325 (b) 3323 (c) 3319 (d) 3320
39. 39.What is the remainder when 59 is divided by 3? (a) 0 (b) 1 (c) 2
40. 40.What is the units digit of 2^20? (a) 8 (b) 6 (c) 2 (d) 4

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