Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 784 · 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 2^23? (a) 2 (b) 4 (c) 6 (d) 8
2. 2.What is the smallest number you must add to 250 to make it divisible by 11? (a) 8 (b) 4 (c) 3 (d) 2
3. 3.What is the remainder when 253 is divided by 12? (a) 0 (b) 2 (c) 1 (d) 11
4. 4.Which of these numbers is divisible by 8? (a) 1122 (b) 1133 (c) 1128 (d) 1140
5. 5.What is the smallest number you must add to 146 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
6. 6.What is the units digit of 3^28? (a) 9 (b) 3 (c) 7 (d) 1
7. 7.What is the smallest number you must add to 498 to make it divisible by 11? (a) 8 (b) 3 (c) 7 (d) 9
8. 8.What is the units digit of 2^47? (a) 2 (b) 6 (c) 8 (d) 4
9. 9.What is the units digit of 3^36? (a) 9 (b) 3 (c) 1 (d) 7
10. 10.Which of these numbers is divisible by 11? (a) 1527 (b) 1517 (c) 1502 (d) 1507
11. 11.What is the units digit of 3^17? (a) 1 (b) 7 (c) 3 (d) 9
12. 12.What is the smallest number you must add to 779 to make it divisible by 11? (a) 9 (b) 3 (c) 1 (d) 2
13. 13.Which of these numbers is divisible by 8? (a) 3120 (b) 3124 (c) 3130 (d) 3114
14. 14.Which of these numbers is divisible by 11? (a) 1384 (b) 1368 (c) 1375 (d) 1388
15. 15.What is the remainder when 196 is divided by 8? (a) 6 (b) 5 (c) 3 (d) 4
16. 16.What is the smallest number you must add to 201 to make it divisible by 4? (a) 1 (b) 3 (c) 2
17. 17.What is the units digit of 8^12? (a) 4 (b) 8 (c) 2 (d) 6
18. 18.Which of these numbers is divisible by 6? (a) 3456 (b) 3467 (c) 3461 (d) 3451
19. 19.What is the units digit of 7^36? (a) 1 (b) 9 (c) 3 (d) 7
20. 20.What is the remainder when 92 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
21. 21.What is the remainder when 676 is divided by 9? (a) 1 (b) 8 (c) 2 (d) 0
22. 22.Which of these numbers is divisible by 8? (a) 5286 (b) 5275 (c) 5271 (d) 5272
23. 23.What is the units digit of 2^40? (a) 6 (b) 4 (c) 8 (d) 2
24. 24.What is the remainder when 145 is divided by 3? (a) 1 (b) 2 (c) 0
25. 25.What is the units digit of 3^51? (a) 3 (b) 9 (c) 1 (d) 7
26. 26.Which of these numbers is divisible by 3? (a) 1403 (b) 1405 (c) 1401 (d) 1400
27. 27.What is the units digit of 4^35? (a) 6 (b) 5 (c) 4 (d) 3
28. 28.Which of these numbers is divisible by 11? (a) 8196 (b) 8184 (c) 8180 (d) 8192
29. 29.What is the remainder when 313 is divided by 6? (a) 2 (b) 5 (c) 1 (d) 0
30. 30.What is the smallest number you must add to 70 to make it divisible by 3? (a) 1 (b) 2
31. 31.What is the units digit of 8^10? (a) 2 (b) 4 (c) 6 (d) 8
32. 32.What is the remainder when 106 is divided by 3? (a) 0 (b) 2 (c) 1
33. 33.Which of these numbers is divisible by 9? (a) 3220 (b) 3208 (c) 3201 (d) 3204
34. 34.What is the remainder when 888 is divided by 11? (a) 9 (b) 3 (c) 8 (d) 7
35. 35.What is the remainder when 184 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
36. 36.Which of these numbers is divisible by 11? (a) 3471 (b) 3449 (c) 3457 (d) 3454
37. 37.What is the units digit of 3^22? (a) 3 (b) 7 (c) 1 (d) 9
38. 38.What is the remainder when 194 is divided by 3? (a) 2 (b) 0 (c) 1
39. 39.What is the units digit of 9^60? (a) 9 (b) 2 (c) 0 (d) 1
40. 40.What is the remainder when 561 is divided by 12? (a) 10 (b) 3 (c) 9 (d) 8

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