Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 494 · 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.Which of these numbers is divisible by 11? (a) 4283 (b) 4295 (c) 4310 (d) 4290
2. 2.What is the remainder when 177 is divided by 7? (a) 5 (b) 2 (c) 3 (d) 1
3. 3.Which of these numbers is divisible by 11? (a) 2431 (b) 2433 (c) 2421 (d) 2447
4. 4.What is the smallest number you must add to 228 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
5. 5.What is the units digit of 4^60? (a) 4 (b) 5 (c) 6 (d) 7
6. 6.What is the units digit of 2^40? (a) 6 (b) 8 (c) 4 (d) 2
7. 7.What is the remainder when 141 is divided by 5? (a) 0 (b) 2 (c) 1 (d) 4
8. 8.What is the smallest number you must add to 146 to make it divisible by 7? (a) 1 (b) 3 (c) 2 (d) 6
9. 9.Which of these numbers is divisible by 4? (a) 1115 (b) 1122 (c) 1116 (d) 1118
10. 10.Which of these numbers is divisible by 8? (a) 789 (b) 807 (c) 796 (d) 792
11. 11.What is the units digit of 8^39? (a) 2 (b) 8 (c) 4 (d) 6
12. 12.Which of these numbers is divisible by 3? (a) 2630 (b) 2627 (c) 2632 (d) 2628
13. 13.What is the units digit of 3^41? (a) 7 (b) 9 (c) 3 (d) 1
14. 14.What is the smallest number you must add to 492 to make it divisible by 7? (a) 2 (b) 4 (c) 5 (d) 6
15. 15.What is the smallest number you must add to 522 to make it divisible by 7? (a) 2 (b) 5 (c) 3 (d) 4
16. 16.What is the smallest number you must add to 62 to make it divisible by 6? (a) 2 (b) 5 (c) 4 (d) 3
17. 17.Which of these numbers is divisible by 6? (a) 4006 (b) 3991 (c) 3996 (d) 4000
18. 18.What is the smallest number you must add to 89 to make it divisible by 4? (a) 3 (b) 1 (c) 2
19. 19.Which of these numbers is divisible by 4? (a) 675 (b) 672 (c) 677 (d) 671
20. 20.Which of these numbers is divisible by 11? (a) 3036 (b) 3051 (c) 3033 (d) 3037
21. 21.What is the remainder when 189 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
22. 22.What is the remainder when 224 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
23. 23.Which of these numbers is divisible by 11? (a) 6865 (b) 6853 (c) 6851 (d) 6855
24. 24.What is the remainder when 76 is divided by 6? (a) 4 (b) 5 (c) 3 (d) 2
25. 25.What is the smallest number you must add to 182 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
26. 26.What is the units digit of 8^29? (a) 6 (b) 4 (c) 2 (d) 8
27. 27.Which of these numbers is divisible by 6? (a) 4898 (b) 4890 (c) 4894 (d) 4888
28. 28.What is the smallest number you must add to 673 to make it divisible by 9? (a) 3 (b) 7 (c) 2 (d) 1
29. 29.What is the remainder when 381 is divided by 7? (a) 5 (b) 4 (c) 2 (d) 3
30. 30.Which of these numbers is divisible by 3? (a) 1308 (b) 1312 (c) 1307 (d) 1310
31. 31.What is the units digit of 9^55? (a) 1 (b) 9 (c) 8 (d) 7
32. 32.What is the smallest number you must add to 157 to make it divisible by 3? (a) 2 (b) 1
33. 33.What is the smallest number you must add to 311 to make it divisible by 4? (a) 1 (b) 3 (c) 2
34. 34.What is the remainder when 61 is divided by 3? (a) 1 (b) 0 (c) 2
35. 35.What is the units digit of 8^35? (a) 4 (b) 6 (c) 2 (d) 8
36. 36.What is the units digit of 8^48? (a) 4 (b) 6 (c) 8 (d) 2
37. 37.What is the remainder when 220 is divided by 6? (a) 4 (b) 3 (c) 2 (d) 5
38. 38.What is the smallest number you must add to 157 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
39. 39.Which of these numbers is divisible by 4? (a) 1566 (b) 1575 (c) 1568 (d) 1569
40. 40.What is the units digit of 8^27? (a) 8 (b) 2 (c) 6 (d) 4

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