Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 975 · 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 4? (a) 2785 (b) 2791 (c) 2782 (d) 2784
2. 2.What is the remainder when 194 is divided by 5? (a) 4 (b) 3 (c) 1 (d) 2
3. 3.What is the smallest number you must add to 244 to make it divisible by 8? (a) 3 (b) 4 (c) 6 (d) 5
4. 4.What is the units digit of 9^55? (a) 1 (b) 8 (c) 9 (d) 7
5. 5.What is the smallest number you must add to 74 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
6. 6.What is the remainder when 509 is divided by 7? (a) 6 (b) 2 (c) 5 (d) 4
7. 7.What is the units digit of 4^18? (a) 5 (b) 6 (c) 7 (d) 4
8. 8.What is the units digit of 3^39? (a) 9 (b) 3 (c) 1 (d) 7
9. 9.What is the remainder when 189 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
10. 10.What is the remainder when 287 is divided by 12? (a) 1 (b) 11 (c) 9 (d) 10
11. 11.What is the units digit of 2^60? (a) 4 (b) 6 (c) 8 (d) 2
12. 12.What is the smallest number you must add to 217 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
13. 13.Which of these numbers is divisible by 3? (a) 2123 (b) 2128 (c) 2126 (d) 2124
14. 14.What is the smallest number you must add to 176 to make it divisible by 7? (a) 4 (b) 6 (c) 5 (d) 1
15. 15.What is the remainder when 294 is divided by 9? (a) 6 (b) 5 (c) 3 (d) 7
16. 16.What is the units digit of 2^35? (a) 8 (b) 4 (c) 6 (d) 2
17. 17.What is the units digit of 8^57? (a) 2 (b) 4 (c) 6 (d) 8
18. 18.What is the units digit of 7^12? (a) 9 (b) 1 (c) 3 (d) 7
19. 19.What is the smallest number you must add to 115 to make it divisible by 3? (a) 2 (b) 1
20. 20.Which of these numbers is divisible by 3? (a) 2468 (b) 2466 (c) 2471 (d) 2465
21. 21.What is the smallest number you must add to 70 to make it divisible by 4? (a) 2 (b) 3 (c) 1
22. 22.What is the smallest number you must add to 124 to make it divisible by 9? (a) 1 (b) 2 (c) 7 (d) 3
23. 23.Which of these numbers is divisible by 9? (a) 4433 (b) 4419 (c) 4411 (d) 4424
24. 24.What is the remainder when 555 is divided by 8? (a) 4 (b) 2 (c) 3 (d) 5
25. 25.What is the units digit of 3^53? (a) 9 (b) 1 (c) 3 (d) 7
26. 26.What is the remainder when 154 is divided by 6? (a) 3 (b) 5 (c) 2 (d) 4
27. 27.What is the remainder when 356 is divided by 5? (a) 4 (b) 0 (c) 1 (d) 2
28. 28.What is the units digit of 9^39? (a) 9 (b) 1 (c) 8 (d) 7
29. 29.What is the remainder when 597 is divided by 9? (a) 4 (b) 2 (c) 6 (d) 3
30. 30.What is the remainder when 577 is divided by 12? (a) 1 (b) 2 (c) 11 (d) 0
31. 31.What is the smallest number you must add to 126 to make it divisible by 10? (a) 5 (b) 4 (c) 3 (d) 6
32. 32.What is the remainder when 169 is divided by 6? (a) 2 (b) 5 (c) 0 (d) 1
33. 33.What is the remainder when 145 is divided by 3? (a) 1 (b) 2 (c) 0
34. 34.Which of these numbers is divisible by 4? (a) 3083 (b) 3087 (c) 3084 (d) 3091
35. 35.What is the smallest number you must add to 237 to make it divisible by 6? (a) 3 (b) 5 (c) 2 (d) 4
36. 36.What is the smallest number you must add to 815 to make it divisible by 11? (a) 9 (b) 1 (c) 10 (d) 8
37. 37.What is the remainder when 212 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
38. 38.What is the units digit of 3^58? (a) 7 (b) 9 (c) 1 (d) 3
39. 39.Which of these numbers is divisible by 9? (a) 1438 (b) 1427 (c) 1431 (d) 1443
40. 40.What is the units digit of 2^19? (a) 6 (b) 8 (c) 4 (d) 2

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