Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 313 · 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 125 to make it divisible by 4? (a) 1 (b) 2 (c) 3
2. 2.What is the remainder when 254 is divided by 5? (a) 3 (b) 4 (c) 2 (d) 1
3. 3.What is the smallest number you must add to 283 to make it divisible by 4? (a) 1 (b) 3 (c) 2
4. 4.Which of these numbers is divisible by 6? (a) 5183 (b) 5167 (c) 5174 (d) 5172
5. 5.What is the units digit of 4^32? (a) 4 (b) 6 (c) 5 (d) 7
6. 6.Which of these numbers is divisible by 3? (a) 1395 (b) 1397 (c) 1393 (d) 1400
7. 7.What is the remainder when 76 is divided by 3? (a) 1 (b) 0 (c) 2
8. 8.What is the units digit of 7^12? (a) 9 (b) 7 (c) 3 (d) 1
9. 9.What is the smallest number you must add to 220 to make it divisible by 3? (a) 1 (b) 2
10. 10.What is the units digit of 8^23? (a) 4 (b) 8 (c) 6 (d) 2
11. 11.What is the units digit of 7^11? (a) 7 (b) 1 (c) 3 (d) 9
12. 12.What is the smallest number you must add to 99 to make it divisible by 6? (a) 3 (b) 2 (c) 4 (d) 5
13. 13.What is the units digit of 8^51? (a) 4 (b) 2 (c) 8 (d) 6
14. 14.Which of these numbers is divisible by 11? (a) 9634 (b) 9645 (c) 9625 (d) 9617
15. 15.Which of these numbers is divisible by 9? (a) 1420 (b) 1422 (c) 1433 (d) 1426
16. 16.What is the smallest number you must add to 425 to make it divisible by 6? (a) 2 (b) 1 (c) 3 (d) 5
17. 17.What is the remainder when 94 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
18. 18.What is the smallest number you must add to 586 to make it divisible by 8? (a) 6 (b) 2 (c) 7 (d) 5
19. 19.What is the smallest number you must add to 847 to make it divisible by 12? (a) 7 (b) 5 (c) 6 (d) 4
20. 20.What is the remainder when 668 is divided by 10? (a) 7 (b) 8 (c) 2 (d) 9
21. 21.What is the remainder when 279 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
22. 22.What is the units digit of 4^56? (a) 4 (b) 6 (c) 7 (d) 5
23. 23.What is the smallest number you must add to 68 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
24. 24.What is the remainder when 191 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
25. 25.What is the remainder when 99 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
26. 26.What is the units digit of 9^23? (a) 7 (b) 9 (c) 1 (d) 8
27. 27.Which of these numbers is divisible by 3? (a) 1074 (b) 1073 (c) 1078 (d) 1076
28. 28.Which of these numbers is divisible by 6? (a) 2283 (b) 2289 (c) 2280 (d) 2275
29. 29.What is the units digit of 7^26? (a) 3 (b) 7 (c) 9 (d) 1
30. 30.What is the units digit of 3^17? (a) 3 (b) 1 (c) 9 (d) 7
31. 31.Which of these numbers is divisible by 3? (a) 149 (b) 146 (c) 151 (d) 147
32. 32.What is the smallest number you must add to 208 to make it divisible by 3? (a) 1 (b) 2
33. 33.What is the remainder when 65 is divided by 6? (a) 1 (b) 4 (c) 3 (d) 5
34. 34.What is the remainder when 62 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
35. 35.What is the smallest number you must add to 87 to make it divisible by 4? (a) 3 (b) 2 (c) 1
36. 36.What is the smallest number you must add to 116 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
37. 37.What is the smallest number you must add to 122 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
38. 38.What is the units digit of 2^12? (a) 4 (b) 2 (c) 8 (d) 6
39. 39.What is the smallest number you must add to 502 to make it divisible by 10? (a) 7 (b) 9 (c) 2 (d) 8
40. 40.Which of these numbers is divisible by 3? (a) 766 (b) 763 (c) 762 (d) 760

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