Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 182 · 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 6? (a) 955 (b) 962 (c) 954 (d) 953
2. 2.What is the smallest number you must add to 675 to make it divisible by 10? (a) 7 (b) 4 (c) 5 (d) 6
3. 3.Which of these numbers is divisible by 11? (a) 8293 (b) 8297 (c) 8309 (d) 8294
4. 4.What is the remainder when 388 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
5. 5.What is the remainder when 357 is divided by 8? (a) 3 (b) 5 (c) 6 (d) 4
6. 6.Which of these numbers is divisible by 4? (a) 1346 (b) 1341 (c) 1339 (d) 1340
7. 7.What is the remainder when 136 is divided by 10? (a) 6 (b) 4 (c) 7 (d) 5
8. 8.What is the units digit of 3^20? (a) 3 (b) 9 (c) 1 (d) 7
9. 9.What is the remainder when 371 is divided by 9? (a) 7 (b) 3 (c) 2 (d) 1
10. 10.What is the units digit of 8^27? (a) 4 (b) 6 (c) 8 (d) 2
11. 11.What is the smallest number you must add to 398 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
12. 12.What is the remainder when 356 is divided by 5? (a) 4 (b) 0 (c) 1 (d) 2
13. 13.What is the smallest number you must add to 136 to make it divisible by 7? (a) 6 (b) 3 (c) 4 (d) 5
14. 14.What is the smallest number you must add to 481 to make it divisible by 6? (a) 1 (b) 4 (c) 5 (d) 3
15. 15.What is the smallest number you must add to 94 to make it divisible by 4? (a) 3 (b) 1 (c) 2
16. 16.What is the smallest number you must add to 407 to make it divisible by 8? (a) 1 (b) 7 (c) 2 (d) 3
17. 17.What is the units digit of 7^33? (a) 3 (b) 7 (c) 9 (d) 1
18. 18.What is the units digit of 8^43? (a) 4 (b) 2 (c) 6 (d) 8
19. 19.Which of these numbers is divisible by 3? (a) 991 (b) 989 (c) 985 (d) 987
20. 20.What is the remainder when 199 is divided by 8? (a) 1 (b) 7 (c) 5 (d) 6
21. 21.What is the smallest number you must add to 291 to make it divisible by 11? (a) 6 (b) 7 (c) 8 (d) 5
22. 22.What is the units digit of 2^15? (a) 8 (b) 4 (c) 2 (d) 6
23. 23.What is the smallest number you must add to 103 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
24. 24.What is the smallest number you must add to 152 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
25. 25.What is the remainder when 225 is divided by 7? (a) 6 (b) 1 (c) 2 (d) 0
26. 26.What is the remainder when 312 is divided by 10? (a) 8 (b) 3 (c) 2 (d) 1
27. 27.Which of these numbers is divisible by 3? (a) 918 (b) 917 (c) 919 (d) 922
28. 28.What is the smallest number you must add to 305 to make it divisible by 8? (a) 1 (b) 6 (c) 7 (d) 5
29. 29.Which of these numbers is divisible by 3? (a) 627 (b) 629 (c) 632 (d) 626
30. 30.What is the units digit of 7^44? (a) 3 (b) 1 (c) 7 (d) 9
31. 31.What is the smallest number you must add to 609 to make it divisible by 11? (a) 4 (b) 6 (c) 8 (d) 7
32. 32.What is the units digit of 2^20? (a) 4 (b) 2 (c) 8 (d) 6
33. 33.What is the smallest number you must add to 598 to make it divisible by 8? (a) 2 (b) 6 (c) 1 (d) 3
34. 34.What is the units digit of 8^41? (a) 4 (b) 8 (c) 6 (d) 2
35. 35.What is the remainder when 154 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
36. 36.What is the units digit of 8^44? (a) 4 (b) 2 (c) 6 (d) 8
37. 37.What is the units digit of 4^28? (a) 7 (b) 5 (c) 4 (d) 6
38. 38.What is the remainder when 254 is divided by 7? (a) 1 (b) 2 (c) 3 (d) 5
39. 39.What is the units digit of 2^49? (a) 4 (b) 6 (c) 8 (d) 2
40. 40.What is the units digit of 3^11? (a) 3 (b) 7 (c) 1 (d) 9

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