Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 539 · 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 units digit of 4^28? (a) 4 (b) 6 (c) 7 (d) 5
2. 2.What is the units digit of 3^40? (a) 1 (b) 9 (c) 3 (d) 7
3. 3.Which of these numbers is divisible by 9? (a) 1036 (b) 1021 (c) 1027 (d) 1026
4. 4.What is the remainder when 128 is divided by 12? (a) 8 (b) 9 (c) 4 (d) 7
5. 5.What is the remainder when 92 is divided by 3? (a) 1 (b) 2 (c) 0
6. 6.Which of these numbers is divisible by 8? (a) 1393 (b) 1392 (c) 1386 (d) 1402
7. 7.What is the remainder when 340 is divided by 9? (a) 6 (b) 8 (c) 7 (d) 2
8. 8.What is the units digit of 3^35? (a) 7 (b) 3 (c) 1 (d) 9
9. 9.What is the units digit of 8^55? (a) 8 (b) 4 (c) 6 (d) 2
10. 10.What is the smallest number you must add to 117 to make it divisible by 4? (a) 2 (b) 1 (c) 3
11. 11.Which of these numbers is divisible by 4? (a) 2315 (b) 2322 (c) 2319 (d) 2316
12. 12.Which of these numbers is divisible by 4? (a) 2296 (b) 2295 (c) 2302 (d) 2299
13. 13.Which of these numbers is divisible by 11? (a) 6444 (b) 6422 (c) 6433 (d) 6424
14. 14.Which of these numbers is divisible by 11? (a) 4895 (b) 4914 (c) 4893 (d) 4899
15. 15.What is the remainder when 549 is divided by 11? (a) 1 (b) 9 (c) 8 (d) 10
16. 16.Which of these numbers is divisible by 6? (a) 4895 (b) 4900 (c) 4890 (d) 4886
17. 17.Which of these numbers is divisible by 3? (a) 1178 (b) 1180 (c) 1176 (d) 1175
18. 18.Which of these numbers is divisible by 3? (a) 1954 (b) 1957 (c) 1953 (d) 1952
19. 19.What is the smallest number you must add to 733 to make it divisible by 12? (a) 9 (b) 11 (c) 10 (d) 1
20. 20.What is the units digit of 3^34? (a) 3 (b) 9 (c) 7 (d) 1
21. 21.What is the units digit of 3^29? (a) 3 (b) 9 (c) 1 (d) 7
22. 22.Which of these numbers is divisible by 8? (a) 5969 (b) 5989 (c) 5980 (d) 5976
23. 23.What is the smallest number you must add to 267 to make it divisible by 6? (a) 4 (b) 2 (c) 3 (d) 5
24. 24.What is the remainder when 125 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
25. 25.What is the remainder when 178 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
26. 26.What is the units digit of 7^50? (a) 9 (b) 7 (c) 3 (d) 1
27. 27.Which of these numbers is divisible by 4? (a) 1926 (b) 1920 (c) 1921 (d) 1917
28. 28.What is the remainder when 512 is divided by 7? (a) 0 (b) 6 (c) 1 (d) 2
29. 29.Which of these numbers is divisible by 8? (a) 5587 (b) 5577 (c) 5598 (d) 5584
30. 30.What is the units digit of 9^17? (a) 7 (b) 9 (c) 1 (d) 8
31. 31.What is the remainder when 907 is divided by 12? (a) 8 (b) 5 (c) 7 (d) 6
32. 32.What is the units digit of 2^52? (a) 2 (b) 4 (c) 6 (d) 8
33. 33.What is the units digit of 3^26? (a) 3 (b) 1 (c) 7 (d) 9
34. 34.What is the smallest number you must add to 303 to make it divisible by 9? (a) 3 (b) 6 (c) 4 (d) 2
35. 35.What is the units digit of 7^54? (a) 9 (b) 1 (c) 3 (d) 7
36. 36.Which of these numbers is divisible by 3? (a) 331 (b) 338 (c) 334 (d) 333
37. 37.What is the smallest number you must add to 785 to make it divisible by 12? (a) 7 (b) 5 (c) 6 (d) 8
38. 38.What is the remainder when 324 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
39. 39.What is the smallest number you must add to 291 to make it divisible by 9? (a) 7 (b) 6 (c) 3 (d) 5
40. 40.What is the remainder when 286 is divided by 5? (a) 0 (b) 2 (c) 1 (d) 4

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad