Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 459 · 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 432 to make it divisible by 7? (a) 3 (b) 5 (c) 1 (d) 2
2. 2.What is the units digit of 3^17? (a) 9 (b) 1 (c) 7 (d) 3
3. 3.What is the units digit of 2^28? (a) 6 (b) 8 (c) 4 (d) 2
4. 4.Which of these numbers is divisible by 3? (a) 1113 (b) 1112 (c) 1115 (d) 1118
5. 5.What is the smallest number you must add to 187 to make it divisible by 9? (a) 1 (b) 7 (c) 2 (d) 3
6. 6.What is the remainder when 855 is divided by 11? (a) 8 (b) 7 (c) 3 (d) 9
7. 7.What is the remainder when 502 is divided by 10? (a) 3 (b) 8 (c) 1 (d) 2
8. 8.Which of these numbers is divisible by 9? (a) 1971 (b) 1963 (c) 1974 (d) 1983
9. 9.Which of these numbers is divisible by 4? (a) 2697 (b) 2700 (c) 2706 (d) 2703
10. 10.What is the remainder when 248 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
11. 11.What is the smallest number you must add to 334 to make it divisible by 11? (a) 4 (b) 6 (c) 8 (d) 7
12. 12.What is the smallest number you must add to 331 to make it divisible by 6? (a) 1 (b) 5 (c) 3 (d) 4
13. 13.What is the smallest number you must add to 344 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
14. 14.What is the remainder when 415 is divided by 6? (a) 0 (b) 1 (c) 2 (d) 5
15. 15.What is the smallest number you must add to 482 to make it divisible by 8? (a) 5 (b) 7 (c) 2 (d) 6
16. 16.What is the remainder when 617 is divided by 8? (a) 2 (b) 0 (c) 1 (d) 7
17. 17.What is the units digit of 4^50? (a) 4 (b) 7 (c) 6 (d) 5
18. 18.Which of these numbers is divisible by 3? (a) 1915 (b) 1918 (c) 1914 (d) 1912
19. 19.Which of these numbers is divisible by 4? (a) 3068 (b) 3073 (c) 3066 (d) 3070
20. 20.What is the units digit of 4^16? (a) 5 (b) 7 (c) 6 (d) 4
21. 21.Which of these numbers is divisible by 3? (a) 1321 (b) 1320 (c) 1325 (d) 1318
22. 22.What is the units digit of 2^50? (a) 8 (b) 2 (c) 6 (d) 4
23. 23.What is the smallest number you must add to 429 to make it divisible by 10? (a) 3 (b) 9 (c) 1 (d) 2
24. 24.Which of these numbers is divisible by 9? (a) 4867 (b) 4884 (c) 4869 (d) 4873
25. 25.What is the smallest number you must add to 529 to make it divisible by 10? (a) 2 (b) 3 (c) 1 (d) 9
26. 26.Which of these numbers is divisible by 4? (a) 2077 (b) 2076 (c) 2083 (d) 2073
27. 27.What is the remainder when 179 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
28. 28.What is the smallest number you must add to 69 to make it divisible by 6? (a) 2 (b) 4 (c) 5 (d) 3
29. 29.What is the smallest number you must add to 202 to make it divisible by 12? (a) 3 (b) 1 (c) 10 (d) 2
30. 30.Which of these numbers is divisible by 11? (a) 2809 (b) 2826 (c) 2837 (d) 2816
31. 31.Which of these numbers is divisible by 11? (a) 5236 (b) 5244 (c) 5234 (d) 5251
32. 32.What is the units digit of 4^42? (a) 7 (b) 6 (c) 5 (d) 4
33. 33.What is the remainder when 53 is divided by 3? (a) 1 (b) 2 (c) 0
34. 34.What is the units digit of 8^32? (a) 2 (b) 8 (c) 4 (d) 6
35. 35.What is the remainder when 218 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
36. 36.Which of these numbers is divisible by 3? (a) 2374 (b) 2378 (c) 2373 (d) 2372
37. 37.What is the units digit of 9^28? (a) 1 (b) 0 (c) 9 (d) 2
38. 38.What is the remainder when 204 is divided by 7? (a) 2 (b) 1 (c) 6 (d) 0
39. 39.What is the remainder when 334 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
40. 40.What is the remainder when 141 is divided by 8? (a) 5 (b) 3 (c) 4 (d) 6

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