Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 743 · 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 344 to make it divisible by 7? (a) 4 (b) 6 (c) 1 (d) 5
2. 2.Which of these numbers is divisible by 4? (a) 305 (b) 314 (c) 310 (d) 308
3. 3.What is the smallest number you must add to 227 to make it divisible by 3? (a) 1 (b) 2
4. 4.What is the remainder when 117 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
5. 5.What is the units digit of 4^45? (a) 4 (b) 6 (c) 3 (d) 5
6. 6.What is the smallest number you must add to 295 to make it divisible by 7? (a) 4 (b) 6 (c) 1 (d) 5
7. 7.Which of these numbers is divisible by 3? (a) 1582 (b) 1577 (c) 1578 (d) 1579
8. 8.What is the units digit of 9^10? (a) 1 (b) 0 (c) 9 (d) 2
9. 9.What is the smallest number you must add to 99 to make it divisible by 7? (a) 5 (b) 1 (c) 4 (d) 6
10. 10.Which of these numbers is divisible by 4? (a) 2550 (b) 2552 (c) 2553 (d) 2558
11. 11.What is the smallest number you must add to 833 to make it divisible by 12? (a) 6 (b) 8 (c) 7 (d) 5
12. 12.What is the units digit of 4^15? (a) 6 (b) 5 (c) 4 (d) 3
13. 13.Which of these numbers is divisible by 9? (a) 1054 (b) 1062 (c) 1074 (d) 1067
14. 14.What is the remainder when 163 is divided by 7? (a) 3 (b) 5 (c) 2 (d) 1
15. 15.What is the remainder when 58 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
16. 16.Which of these numbers is divisible by 4? (a) 2944 (b) 2946 (c) 2943 (d) 2949
17. 17.What is the units digit of 2^50? (a) 4 (b) 8 (c) 6 (d) 2
18. 18.What is the remainder when 86 is divided by 7? (a) 1 (b) 2 (c) 5 (d) 3
19. 19.Which of these numbers is divisible by 4? (a) 340 (b) 342 (c) 345 (d) 338
20. 20.What is the remainder when 413 is divided by 6? (a) 3 (b) 1 (c) 5 (d) 4
21. 21.What is the units digit of 3^17? (a) 3 (b) 9 (c) 7 (d) 1
22. 22.What is the smallest number you must add to 298 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 2
23. 23.Which of these numbers is divisible by 6? (a) 966 (b) 970 (c) 977 (d) 964
24. 24.What is the remainder when 386 is divided by 7? (a) 6 (b) 2 (c) 1 (d) 0
25. 25.What is the units digit of 8^54? (a) 8 (b) 2 (c) 6 (d) 4
26. 26.What is the units digit of 9^58? (a) 2 (b) 0 (c) 1 (d) 9
27. 27.What is the units digit of 2^18? (a) 6 (b) 4 (c) 2 (d) 8
28. 28.What is the smallest number you must add to 191 to make it divisible by 3? (a) 1 (b) 2
29. 29.What is the smallest number you must add to 43 to make it divisible by 4? (a) 2 (b) 3 (c) 1
30. 30.What is the remainder when 181 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
31. 31.What is the remainder when 50 is divided by 3? (a) 2 (b) 1 (c) 0
32. 32.What is the smallest number you must add to 904 to make it divisible by 12? (a) 7 (b) 9 (c) 8 (d) 4
33. 33.What is the units digit of 8^13? (a) 4 (b) 6 (c) 2 (d) 8
34. 34.What is the smallest number you must add to 957 to make it divisible by 12? (a) 3 (b) 4 (c) 9 (d) 2
35. 35.What is the remainder when 94 is divided by 9? (a) 5 (b) 4 (c) 6 (d) 3
36. 36.What is the units digit of 4^26? (a) 6 (b) 7 (c) 5 (d) 4
37. 37.Which of these numbers is divisible by 8? (a) 4891 (b) 4896 (c) 4898 (d) 4908
38. 38.What is the remainder when 502 is divided by 7? (a) 2 (b) 6 (c) 4 (d) 5
39. 39.Which of these numbers is divisible by 6? (a) 1966 (b) 1973 (c) 1975 (d) 1968
40. 40.What is the remainder when 363 is divided by 12? (a) 9 (b) 3 (c) 2 (d) 4

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