Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 826 · 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 9? (a) 458 (b) 466 (c) 475 (d) 459
2. 2.What is the units digit of 4^22? (a) 4 (b) 7 (c) 5 (d) 6
3. 3.What is the remainder when 254 is divided by 7? (a) 5 (b) 1 (c) 3 (d) 2
4. 4.What is the units digit of 4^58? (a) 5 (b) 4 (c) 7 (d) 6
5. 5.What is the smallest number you must add to 68 to make it divisible by 6? (a) 2 (b) 5 (c) 4 (d) 3
6. 6.What is the units digit of 8^11? (a) 2 (b) 6 (c) 4 (d) 8
7. 7.What is the smallest number you must add to 722 to make it divisible by 11? (a) 5 (b) 7 (c) 3 (d) 4
8. 8.What is the units digit of 4^19? (a) 3 (b) 5 (c) 6 (d) 4
9. 9.What is the remainder when 146 is divided by 5? (a) 2 (b) 0 (c) 4 (d) 1
10. 10.What is the remainder when 208 is divided by 3? (a) 1 (b) 0 (c) 2
11. 11.What is the units digit of 3^41? (a) 1 (b) 9 (c) 3 (d) 7
12. 12.What is the smallest number you must add to 393 to make it divisible by 12? (a) 2 (b) 4 (c) 9 (d) 3
13. 13.What is the units digit of 7^36? (a) 9 (b) 3 (c) 7 (d) 1
14. 14.What is the remainder when 227 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
15. 15.What is the smallest number you must add to 94 to make it divisible by 6? (a) 1 (b) 3 (c) 4 (d) 2
16. 16.What is the units digit of 8^22? (a) 6 (b) 4 (c) 2 (d) 8
17. 17.What is the remainder when 227 is divided by 10? (a) 6 (b) 8 (c) 3 (d) 7
18. 18.Which of these numbers is divisible by 11? (a) 3970 (b) 3972 (c) 3971 (d) 3990
19. 19.What is the remainder when 195 is divided by 6? (a) 5 (b) 3 (c) 4 (d) 2
20. 20.What is the smallest number you must add to 637 to make it divisible by 10? (a) 7 (b) 4 (c) 2 (d) 3
21. 21.Which of these numbers is divisible by 11? (a) 2174 (b) 2178 (c) 2198 (d) 2188
22. 22.What is the remainder when 271 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 0
23. 23.What is the units digit of 3^43? (a) 3 (b) 9 (c) 1 (d) 7
24. 24.What is the smallest number you must add to 151 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
25. 25.What is the remainder when 682 is divided by 10? (a) 1 (b) 3 (c) 2 (d) 8
26. 26.What is the smallest number you must add to 107 to make it divisible by 3? (a) 1 (b) 2
27. 27.What is the units digit of 9^40? (a) 0 (b) 9 (c) 1 (d) 2
28. 28.What is the remainder when 142 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
29. 29.What is the remainder when 475 is divided by 11? (a) 1 (b) 9 (c) 3 (d) 2
30. 30.Which of these numbers is divisible by 9? (a) 6041 (b) 6037 (c) 6049 (d) 6039
31. 31.What is the units digit of 4^28? (a) 5 (b) 6 (c) 4 (d) 7
32. 32.Which of these numbers is divisible by 6? (a) 2554 (b) 2548 (c) 2550 (d) 2558
33. 33.Which of these numbers is divisible by 9? (a) 1960 (b) 1963 (c) 1975 (d) 1962
34. 34.What is the remainder when 187 is divided by 9? (a) 8 (b) 6 (c) 2 (d) 7
35. 35.What is the smallest number you must add to 476 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
36. 36.Which of these numbers is divisible by 6? (a) 2982 (b) 2981 (c) 2986 (d) 2989
37. 37.What is the smallest number you must add to 291 to make it divisible by 4? (a) 3 (b) 1 (c) 2
38. 38.Which of these numbers is divisible by 6? (a) 678 (b) 680 (c) 673 (d) 687
39. 39.What is the smallest number you must add to 620 to make it divisible by 11? (a) 7 (b) 6 (c) 4 (d) 8
40. 40.What is the smallest number you must add to 463 to make it divisible by 6? (a) 3 (b) 4 (c) 1 (d) 5

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