Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 375 · 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 504 to make it divisible by 10? (a) 6 (b) 4 (c) 5 (d) 7
2. 2.What is the smallest number you must add to 86 to make it divisible by 8? (a) 3 (b) 6 (c) 1 (d) 2
3. 3.What is the remainder when 470 is divided by 8? (a) 5 (b) 7 (c) 2 (d) 6
4. 4.What is the units digit of 7^11? (a) 9 (b) 7 (c) 3 (d) 1
5. 5.What is the remainder when 184 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
6. 6.What is the units digit of 4^11? (a) 5 (b) 4 (c) 3 (d) 6
7. 7.Which of these numbers is divisible by 11? (a) 2024 (b) 2036 (c) 2026 (d) 2020
8. 8.What is the smallest number you must add to 162 to make it divisible by 7? (a) 1 (b) 6 (c) 5 (d) 4
9. 9.Which of these numbers is divisible by 9? (a) 2472 (b) 2466 (c) 2462 (d) 2482
10. 10.What is the remainder when 708 is divided by 10? (a) 2 (b) 9 (c) 7 (d) 8
11. 11.What is the remainder when 432 is divided by 10? (a) 8 (b) 3 (c) 1 (d) 2
12. 12.What is the remainder when 218 is divided by 5? (a) 4 (b) 1 (c) 3 (d) 2
13. 13.What is the remainder when 193 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
14. 14.What is the units digit of 3^55? (a) 3 (b) 9 (c) 1 (d) 7
15. 15.What is the smallest number you must add to 424 to make it divisible by 9? (a) 8 (b) 7 (c) 6 (d) 1
16. 16.What is the remainder when 159 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
17. 17.What is the units digit of 7^27? (a) 3 (b) 1 (c) 7 (d) 9
18. 18.Which of these numbers is divisible by 8? (a) 4444 (b) 4463 (c) 4451 (d) 4448
19. 19.What is the remainder when 348 is divided by 10? (a) 7 (b) 8 (c) 9 (d) 2
20. 20.What is the smallest number you must add to 332 to make it divisible by 11? (a) 2 (b) 9 (c) 8 (d) 10
21. 21.Which of these numbers is divisible by 6? (a) 3208 (b) 3218 (c) 3215 (d) 3210
22. 22.Which of these numbers is divisible by 3? (a) 2577 (b) 2581 (c) 2576 (d) 2578
23. 23.What is the units digit of 3^12? (a) 3 (b) 7 (c) 1 (d) 9
24. 24.What is the smallest number you must add to 313 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
25. 25.What is the smallest number you must add to 189 to make it divisible by 6? (a) 2 (b) 4 (c) 5 (d) 3
26. 26.What is the units digit of 7^41? (a) 9 (b) 7 (c) 3 (d) 1
27. 27.What is the remainder when 53 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
28. 28.What is the remainder when 124 is divided by 3? (a) 1 (b) 2 (c) 0
29. 29.What is the remainder when 109 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
30. 30.What is the smallest number you must add to 652 to make it divisible by 11? (a) 7 (b) 8 (c) 9 (d) 3
31. 31.What is the units digit of 9^53? (a) 1 (b) 9 (c) 7 (d) 8
32. 32.What is the smallest number you must add to 360 to make it divisible by 7? (a) 3 (b) 6 (c) 4 (d) 5
33. 33.What is the smallest number you must add to 245 to make it divisible by 9? (a) 2 (b) 7 (c) 6 (d) 8
34. 34.What is the units digit of 8^53? (a) 8 (b) 6 (c) 2 (d) 4
35. 35.Which of these numbers is divisible by 11? (a) 3870 (b) 3888 (c) 3878 (d) 3872
36. 36.What is the smallest number you must add to 800 to make it divisible by 11? (a) 2 (b) 8 (c) 4 (d) 3
37. 37.What is the remainder when 209 is divided by 3? (a) 0 (b) 2 (c) 1
38. 38.Which of these numbers is divisible by 11? (a) 3674 (b) 3675 (c) 3668 (d) 3691
39. 39.Which of these numbers is divisible by 4? (a) 2039 (b) 2042 (c) 2046 (d) 2040
40. 40.What is the remainder when 103 is divided by 3? (a) 2 (b) 1 (c) 0

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