Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 123 · 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 288 to make it divisible by 11? (a) 9 (b) 2 (c) 10 (d) 8
2. 2.Which of these numbers is divisible by 9? (a) 2178 (b) 2194 (c) 2179 (d) 2175
3. 3.What is the remainder when 143 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
4. 4.What is the units digit of 3^21? (a) 1 (b) 3 (c) 7 (d) 9
5. 5.What is the remainder when 215 is divided by 6? (a) 1 (b) 4 (c) 5 (d) 3
6. 6.What is the remainder when 114 is divided by 9? (a) 3 (b) 5 (c) 6 (d) 7
7. 7.What is the smallest number you must add to 678 to make it divisible by 12? (a) 6 (b) 8 (c) 7 (d) 5
8. 8.What is the units digit of 4^55? (a) 5 (b) 6 (c) 4 (d) 3
9. 9.What is the units digit of 8^25? (a) 2 (b) 8 (c) 4 (d) 6
10. 10.What is the remainder when 278 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
11. 11.What is the smallest number you must add to 117 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
12. 12.What is the units digit of 4^50? (a) 4 (b) 5 (c) 7 (d) 6
13. 13.Which of these numbers is divisible by 9? (a) 6412 (b) 6428 (c) 6417 (d) 6418
14. 14.What is the smallest number you must add to 233 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
15. 15.What is the smallest number you must add to 49 to make it divisible by 3? (a) 1 (b) 2
16. 16.What is the smallest number you must add to 207 to make it divisible by 11? (a) 9 (b) 1 (c) 2 (d) 3
17. 17.What is the smallest number you must add to 319 to make it divisible by 4? (a) 2 (b) 3 (c) 1
18. 18.What is the remainder when 612 is divided by 10? (a) 1 (b) 3 (c) 2 (d) 8
19. 19.What is the remainder when 567 is divided by 8? (a) 1 (b) 5 (c) 7 (d) 6
20. 20.What is the remainder when 274 is divided by 10? (a) 3 (b) 4 (c) 6 (d) 5
21. 21.Which of these numbers is divisible by 8? (a) 1224 (b) 1218 (c) 1234 (d) 1225
22. 22.What is the units digit of 8^48? (a) 2 (b) 6 (c) 8 (d) 4
23. 23.Which of these numbers is divisible by 9? (a) 2818 (b) 2817 (c) 2832 (d) 2816
24. 24.What is the remainder when 44 is divided by 3? (a) 0 (b) 2 (c) 1
25. 25.Which of these numbers is divisible by 11? (a) 9052 (b) 9042 (c) 9054 (d) 9034
26. 26.What is the smallest number you must add to 273 to make it divisible by 12? (a) 4 (b) 9 (c) 2 (d) 3
27. 27.Which of these numbers is divisible by 9? (a) 386 (b) 402 (c) 387 (d) 393
28. 28.What is the smallest number you must add to 357 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
29. 29.What is the units digit of 4^29? (a) 5 (b) 4 (c) 6 (d) 3
30. 30.What is the units digit of 4^58? (a) 6 (b) 7 (c) 5 (d) 4
31. 31.What is the units digit of 8^60? (a) 4 (b) 2 (c) 6 (d) 8
32. 32.What is the smallest number you must add to 543 to make it divisible by 12? (a) 9 (b) 8 (c) 10 (d) 3
33. 33.Which of these numbers is divisible by 3? (a) 1867 (b) 1873 (c) 1871 (d) 1869
34. 34.What is the smallest number you must add to 152 to make it divisible by 6? (a) 3 (b) 4 (c) 2 (d) 5
35. 35.What is the remainder when 248 is divided by 10? (a) 9 (b) 2 (c) 7 (d) 8
36. 36.What is the smallest number you must add to 476 to make it divisible by 9? (a) 2 (b) 8 (c) 1 (d) 3
37. 37.What is the remainder when 445 is divided by 11? (a) 5 (b) 4 (c) 6 (d) 7
38. 38.Which of these numbers is divisible by 8? (a) 5248 (b) 5261 (c) 5253 (d) 5246
39. 39.What is the units digit of 9^20? (a) 0 (b) 2 (c) 9 (d) 1
40. 40.Which of these numbers is divisible by 11? (a) 4420 (b) 4402 (c) 4400 (d) 4391

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