Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 46 · 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 9? (a) 2804 (b) 2793 (c) 2790 (d) 2786
2. 2.What is the units digit of 4^30? (a) 5 (b) 4 (c) 6 (d) 7
3. 3.Which of these numbers is divisible by 4? (a) 2401 (b) 2400 (c) 2405 (d) 2397
4. 4.Which of these numbers is divisible by 3? (a) 1001 (b) 996 (c) 995 (d) 997
5. 5.What is the smallest number you must add to 352 to make it divisible by 6? (a) 3 (b) 4 (c) 1 (d) 2
6. 6.What is the remainder when 802 is divided by 10? (a) 2 (b) 1 (c) 8 (d) 3
7. 7.What is the units digit of 8^39? (a) 2 (b) 6 (c) 4 (d) 8
8. 8.What is the units digit of 2^38? (a) 2 (b) 4 (c) 6 (d) 8
9. 9.What is the smallest number you must add to 169 to make it divisible by 7? (a) 6 (b) 1 (c) 5 (d) 4
10. 10.What is the remainder when 218 is divided by 10? (a) 9 (b) 2 (c) 8 (d) 7
11. 11.What is the remainder when 496 is divided by 11? (a) 0 (b) 1 (c) 10 (d) 2
12. 12.What is the smallest number you must add to 590 to make it divisible by 8? (a) 2 (b) 1 (c) 3 (d) 6
13. 13.What is the smallest number you must add to 639 to make it divisible by 8? (a) 7 (b) 2 (c) 3 (d) 1
14. 14.What is the remainder when 385 is divided by 6? (a) 2 (b) 1 (c) 0 (d) 5
15. 15.Which of these numbers is divisible by 4? (a) 2362 (b) 2360 (c) 2357 (d) 2365
16. 16.What is the remainder when 343 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
17. 17.Which of these numbers is divisible by 3? (a) 370 (b) 373 (c) 368 (d) 369
18. 18.Which of these numbers is divisible by 9? (a) 825 (b) 843 (c) 831 (d) 828
19. 19.What is the units digit of 2^55? (a) 8 (b) 6 (c) 2 (d) 4
20. 20.Which of these numbers is divisible by 4? (a) 2994 (b) 2985 (c) 2988 (d) 2990
21. 21.What is the units digit of 4^18? (a) 7 (b) 6 (c) 4 (d) 5
22. 22.What is the units digit of 8^32? (a) 4 (b) 2 (c) 8 (d) 6
23. 23.What is the smallest number you must add to 275 to make it divisible by 10? (a) 7 (b) 5 (c) 6 (d) 4
24. 24.What is the remainder when 186 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
25. 25.What is the units digit of 2^21? (a) 8 (b) 4 (c) 6 (d) 2
26. 26.What is the units digit of 8^15? (a) 8 (b) 2 (c) 4 (d) 6
27. 27.What is the units digit of 2^24? (a) 4 (b) 6 (c) 8 (d) 2
28. 28.Which of these numbers is divisible by 6? (a) 1914 (b) 1916 (c) 1909 (d) 1923
29. 29.What is the smallest number you must add to 604 to make it divisible by 9? (a) 7 (b) 8 (c) 1 (d) 6
30. 30.Which of these numbers is divisible by 4? (a) 1381 (b) 1380 (c) 1386 (d) 1378
31. 31.What is the units digit of 4^36? (a) 7 (b) 5 (c) 4 (d) 6
32. 32.What is the remainder when 543 is divided by 9? (a) 6 (b) 2 (c) 3 (d) 4
33. 33.Which of these numbers is divisible by 8? (a) 1755 (b) 1740 (c) 1744 (d) 1747
34. 34.What is the smallest number you must add to 401 to make it divisible by 11? (a) 7 (b) 5 (c) 6 (d) 8
35. 35.What is the remainder when 154 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
36. 36.What is the smallest number you must add to 430 to make it divisible by 7? (a) 5 (b) 6 (c) 4 (d) 3
37. 37.Which of these numbers is divisible by 9? (a) 4710 (b) 4716 (c) 4724 (d) 4731
38. 38.What is the units digit of 3^37? (a) 1 (b) 9 (c) 7 (d) 3
39. 39.What is the smallest number you must add to 289 to make it divisible by 4? (a) 1 (b) 2 (c) 3
40. 40.What is the remainder when 293 is divided by 6? (a) 3 (b) 4 (c) 1 (d) 5

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