Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 431 · 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 157 to make it divisible by 3? (a) 2 (b) 1
2. 2.What is the units digit of 8^38? (a) 4 (b) 6 (c) 2 (d) 8
3. 3.What is the smallest number you must add to 202 to make it divisible by 3? (a) 2 (b) 1
4. 4.Which of these numbers is divisible by 8? (a) 3232 (b) 3244 (c) 3230 (d) 3236
5. 5.What is the smallest number you must add to 238 to make it divisible by 6? (a) 1 (b) 3 (c) 2 (d) 4
6. 6.What is the remainder when 481 is divided by 8? (a) 7 (b) 1 (c) 2 (d) 0
7. 7.Which of these numbers is divisible by 9? (a) 4220 (b) 4212 (c) 4228 (d) 4211
8. 8.What is the remainder when 191 is divided by 6? (a) 4 (b) 3 (c) 1 (d) 5
9. 9.What is the units digit of 2^31? (a) 8 (b) 6 (c) 2 (d) 4
10. 10.What is the remainder when 194 is divided by 3? (a) 1 (b) 0 (c) 2
11. 11.Which of these numbers is divisible by 4? (a) 2175 (b) 2170 (c) 2168 (d) 2165
12. 12.What is the units digit of 7^34? (a) 3 (b) 1 (c) 7 (d) 9
13. 13.What is the remainder when 236 is divided by 6? (a) 1 (b) 2 (c) 4 (d) 3
14. 14.Which of these numbers is divisible by 3? (a) 305 (b) 303 (c) 307 (d) 302
15. 15.What is the remainder when 691 is divided by 10? (a) 0 (b) 2 (c) 9 (d) 1
16. 16.What is the smallest number you must add to 145 to make it divisible by 3? (a) 1 (b) 2
17. 17.Which of these numbers is divisible by 9? (a) 3655 (b) 3674 (c) 3663 (d) 3669
18. 18.What is the remainder when 460 is divided by 9? (a) 1 (b) 8 (c) 2 (d) 0
19. 19.What is the smallest number you must add to 633 to make it divisible by 10? (a) 3 (b) 8 (c) 6 (d) 7
20. 20.Which of these numbers is divisible by 3? (a) 1663 (b) 1659 (c) 1660 (d) 1657
21. 21.What is the remainder when 284 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
22. 22.What is the units digit of 7^14? (a) 9 (b) 7 (c) 3 (d) 1
23. 23.What is the smallest number you must add to 395 to make it divisible by 7? (a) 4 (b) 5 (c) 3 (d) 6
24. 24.Which of these numbers is divisible by 9? (a) 1042 (b) 1054 (c) 1047 (d) 1044
25. 25.What is the units digit of 9^41? (a) 8 (b) 1 (c) 9 (d) 7
26. 26.Which of these numbers is divisible by 11? (a) 3180 (b) 3179 (c) 3193 (d) 3178
27. 27.What is the remainder when 286 is divided by 8? (a) 5 (b) 7 (c) 6 (d) 2
28. 28.What is the smallest number you must add to 105 to make it divisible by 8? (a) 6 (b) 1 (c) 7 (d) 5
29. 29.What is the smallest number you must add to 233 to make it divisible by 3? (a) 2 (b) 1
30. 30.What is the units digit of 2^39? (a) 6 (b) 4 (c) 2 (d) 8
31. 31.What is the smallest number you must add to 229 to make it divisible by 7? (a) 2 (b) 5 (c) 1 (d) 3
32. 32.What is the remainder when 505 is divided by 10? (a) 5 (b) 6 (c) 7 (d) 4
33. 33.What is the remainder when 606 is divided by 10? (a) 7 (b) 4 (c) 6 (d) 5
34. 34.What is the smallest number you must add to 752 to make it divisible by 11? (a) 8 (b) 4 (c) 6 (d) 7
35. 35.Which of these numbers is divisible by 9? (a) 2199 (b) 2181 (c) 2187 (d) 2189
36. 36.Which of these numbers is divisible by 8? (a) 500 (b) 486 (c) 492 (d) 488
37. 37.What is the remainder when 445 is divided by 12? (a) 0 (b) 1 (c) 11 (d) 2
38. 38.What is the smallest number you must add to 166 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
39. 39.What is the units digit of 8^14? (a) 8 (b) 4 (c) 6 (d) 2
40. 40.What is the remainder when 104 is divided by 3? (a) 2 (b) 1 (c) 0

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