Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 135 · 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 remainder when 174 is divided by 11? (a) 10 (b) 2 (c) 9 (d) 8
2. 2.What is the smallest number you must add to 151 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
3. 3.Which of these numbers is divisible by 4? (a) 887 (b) 889 (c) 881 (d) 884
4. 4.Which of these numbers is divisible by 11? (a) 7172 (b) 7182 (c) 7164 (d) 7184
5. 5.What is the smallest number you must add to 637 to make it divisible by 11? (a) 2 (b) 1 (c) 10 (d) 3
6. 6.What is the remainder when 114 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
7. 7.What is the units digit of 3^44? (a) 1 (b) 7 (c) 9 (d) 3
8. 8.What is the smallest number you must add to 139 to make it divisible by 7? (a) 1 (b) 3 (c) 2 (d) 6
9. 9.What is the remainder when 208 is divided by 3? (a) 2 (b) 0 (c) 1
10. 10.What is the smallest number you must add to 288 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
11. 11.Which of these numbers is divisible by 3? (a) 1449 (b) 1450 (c) 1454 (d) 1447
12. 12.What is the smallest number you must add to 109 to make it divisible by 9? (a) 8 (b) 7 (c) 6 (d) 1
13. 13.What is the units digit of 2^18? (a) 4 (b) 6 (c) 2 (d) 8
14. 14.What is the smallest number you must add to 314 to make it divisible by 4? (a) 1 (b) 3 (c) 2
15. 15.What is the remainder when 133 is divided by 6? (a) 1 (b) 2 (c) 5 (d) 0
16. 16.What is the units digit of 7^46? (a) 9 (b) 3 (c) 7 (d) 1
17. 17.What is the remainder when 220 is divided by 3? (a) 0 (b) 1 (c) 2
18. 18.What is the smallest number you must add to 376 to make it divisible by 9? (a) 1 (b) 2 (c) 7 (d) 3
19. 19.What is the units digit of 3^13? (a) 7 (b) 9 (c) 3 (d) 1
20. 20.Which of these numbers is divisible by 4? (a) 2480 (b) 2482 (c) 2486 (d) 2478
21. 21.Which of these numbers is divisible by 9? (a) 3499 (b) 3513 (c) 3501 (d) 3502
22. 22.What is the remainder when 541 is divided by 9? (a) 1 (b) 0 (c) 8 (d) 2
23. 23.What is the units digit of 2^50? (a) 4 (b) 8 (c) 6 (d) 2
24. 24.What is the smallest number you must add to 68 to make it divisible by 3? (a) 1 (b) 2
25. 25.What is the smallest number you must add to 157 to make it divisible by 3? (a) 2 (b) 1
26. 26.Which of these numbers is divisible by 11? (a) 734 (b) 737 (c) 758 (d) 740
27. 27.What is the remainder when 259 is divided by 6? (a) 1 (b) 0 (c) 5 (d) 2
28. 28.What is the smallest number you must add to 116 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
29. 29.What is the remainder when 282 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
30. 30.Which of these numbers is divisible by 9? (a) 5184 (b) 5189 (c) 5201 (d) 5180
31. 31.Which of these numbers is divisible by 8? (a) 3029 (b) 3037 (c) 3024 (d) 3019
32. 32.What is the units digit of 4^49? (a) 4 (b) 5 (c) 3 (d) 6
33. 33.What is the units digit of 9^31? (a) 1 (b) 8 (c) 9 (d) 7
34. 34.Which of these numbers is divisible by 11? (a) 5004 (b) 4983 (c) 4992 (d) 4979
35. 35.What is the units digit of 9^50? (a) 0 (b) 1 (c) 9 (d) 2
36. 36.What is the remainder when 630 is divided by 8? (a) 7 (b) 6 (c) 5 (d) 2
37. 37.What is the units digit of 7^59? (a) 1 (b) 3 (c) 7 (d) 9
38. 38.What is the units digit of 3^46? (a) 9 (b) 7 (c) 3 (d) 1
39. 39.Which of these numbers is divisible by 4? (a) 1251 (b) 1244 (c) 1247 (d) 1241
40. 40.What is the remainder when 150 is divided by 7? (a) 2 (b) 4 (c) 5 (d) 3

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