Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 286 · 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 744 to make it divisible by 11? (a) 5 (b) 7 (c) 4 (d) 3
2. 2.What is the units digit of 8^46? (a) 6 (b) 4 (c) 8 (d) 2
3. 3.What is the smallest number you must add to 222 to make it divisible by 11? (a) 2 (b) 9 (c) 10 (d) 8
4. 4.Which of these numbers is divisible by 11? (a) 4534 (b) 4543 (c) 4551 (d) 4564
5. 5.What is the units digit of 9^42? (a) 2 (b) 1 (c) 0 (d) 9
6. 6.Which of these numbers is divisible by 3? (a) 2145 (b) 2143 (c) 2147 (d) 2149
7. 7.Which of these numbers is divisible by 6? (a) 4994 (b) 5002 (c) 4988 (d) 4992
8. 8.What is the remainder when 264 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
9. 9.Which of these numbers is divisible by 11? (a) 2417 (b) 2400 (c) 2398 (d) 2395
10. 10.What is the smallest number you must add to 83 to make it divisible by 3? (a) 2 (b) 1
11. 11.What is the remainder when 235 is divided by 3? (a) 1 (b) 0 (c) 2
12. 12.Which of these numbers is divisible by 8? (a) 407 (b) 417 (c) 408 (d) 411
13. 13.What is the remainder when 175 is divided by 3? (a) 0 (b) 2 (c) 1
14. 14.Which of these numbers is divisible by 3? (a) 910 (b) 904 (c) 906 (d) 908
15. 15.What is the remainder when 261 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
16. 16.What is the smallest number you must add to 554 to make it divisible by 11? (a) 7 (b) 8 (c) 6 (d) 4
17. 17.Which of these numbers is divisible by 6? (a) 4153 (b) 4161 (c) 4158 (d) 4168
18. 18.Which of these numbers is divisible by 9? (a) 2350 (b) 2365 (c) 2345 (d) 2349
19. 19.Which of these numbers is divisible by 6? (a) 1391 (b) 1380 (c) 1381 (d) 1379
20. 20.What is the smallest number you must add to 245 to make it divisible by 4? (a) 3 (b) 1 (c) 2
21. 21.What is the units digit of 2^33? (a) 4 (b) 2 (c) 6 (d) 8
22. 22.What is the units digit of 7^49? (a) 3 (b) 1 (c) 7 (d) 9
23. 23.Which of these numbers is divisible by 3? (a) 557 (b) 559 (c) 555 (d) 554
24. 24.Which of these numbers is divisible by 3? (a) 2026 (b) 2028 (c) 2030 (d) 2032
25. 25.Which of these numbers is divisible by 4? (a) 1489 (b) 1486 (c) 1488 (d) 1495
26. 26.Which of these numbers is divisible by 8? (a) 342 (b) 336 (c) 347 (d) 333
27. 27.Which of these numbers is divisible by 9? (a) 809 (b) 792 (c) 790 (d) 796
28. 28.What is the smallest number you must add to 187 to make it divisible by 4? (a) 3 (b) 1 (c) 2
29. 29.What is the remainder when 860 is divided by 11? (a) 3 (b) 9 (c) 1 (d) 2
30. 30.What is the remainder when 329 is divided by 12? (a) 5 (b) 6 (c) 7 (d) 4
31. 31.Which of these numbers is divisible by 3? (a) 2150 (b) 2147 (c) 2152 (d) 2148
32. 32.What is the remainder when 226 is divided by 8? (a) 6 (b) 2 (c) 1 (d) 3
33. 33.What is the units digit of 4^22? (a) 7 (b) 4 (c) 5 (d) 6
34. 34.What is the remainder when 237 is divided by 8? (a) 4 (b) 6 (c) 3 (d) 5
35. 35.Which of these numbers is divisible by 9? (a) 6567 (b) 6573 (c) 6559 (d) 6561
36. 36.Which of these numbers is divisible by 11? (a) 1836 (b) 1839 (c) 1837 (d) 1849
37. 37.What is the remainder when 233 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
38. 38.Which of these numbers is divisible by 9? (a) 6198 (b) 6192 (c) 6209 (d) 6185
39. 39.Which of these numbers is divisible by 8? (a) 3533 (b) 3522 (c) 3518 (d) 3520
40. 40.What is the remainder when 163 is divided by 3? (a) 2 (b) 1 (c) 0

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