Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 929 · 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 456 is divided by 9? (a) 5 (b) 3 (c) 7 (d) 6
2. 2.Which of these numbers is divisible by 8? (a) 354 (b) 351 (c) 343 (d) 344
3. 3.Which of these numbers is divisible by 4? (a) 2768 (b) 2767 (c) 2775 (d) 2769
4. 4.What is the smallest number you must add to 230 to make it divisible by 11? (a) 10 (b) 2 (c) 3 (d) 1
5. 5.What is the remainder when 55 is divided by 3? (a) 2 (b) 1 (c) 0
6. 6.What is the units digit of 9^42? (a) 2 (b) 9 (c) 0 (d) 1
7. 7.What is the remainder when 773 is divided by 12? (a) 4 (b) 7 (c) 6 (d) 5
8. 8.What is the units digit of 4^20? (a) 7 (b) 6 (c) 5 (d) 4
9. 9.What is the smallest number you must add to 446 to make it divisible by 11? (a) 7 (b) 6 (c) 4 (d) 5
10. 10.What is the remainder when 83 is divided by 3? (a) 1 (b) 0 (c) 2
11. 11.Which of these numbers is divisible by 6? (a) 961 (b) 967 (c) 958 (d) 960
12. 12.What is the units digit of 4^16? (a) 5 (b) 4 (c) 6 (d) 7
13. 13.Which of these numbers is divisible by 6? (a) 1301 (b) 1307 (c) 1296 (d) 1295
14. 14.What is the remainder when 512 is divided by 10? (a) 8 (b) 1 (c) 2 (d) 3
15. 15.What is the units digit of 4^14? (a) 5 (b) 7 (c) 4 (d) 6
16. 16.What is the remainder when 254 is divided by 12? (a) 10 (b) 1 (c) 3 (d) 2
17. 17.Which of these numbers is divisible by 6? (a) 4277 (b) 4266 (c) 4267 (d) 4264
18. 18.What is the units digit of 9^27? (a) 7 (b) 8 (c) 9 (d) 1
19. 19.What is the remainder when 68 is divided by 3? (a) 2 (b) 1 (c) 0
20. 20.What is the smallest number you must add to 166 to make it divisible by 4? (a) 2 (b) 1 (c) 3
21. 21.What is the smallest number you must add to 174 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
22. 22.What is the smallest number you must add to 92 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 2
23. 23.What is the remainder when 97 is divided by 6? (a) 0 (b) 5 (c) 2 (d) 1
24. 24.What is the units digit of 8^23? (a) 6 (b) 4 (c) 2 (d) 8
25. 25.What is the smallest number you must add to 267 to make it divisible by 4? (a) 3 (b) 1 (c) 2
26. 26.What is the units digit of 7^54? (a) 9 (b) 3 (c) 1 (d) 7
27. 27.What is the smallest number you must add to 157 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 1
28. 28.What is the units digit of 2^45? (a) 4 (b) 8 (c) 6 (d) 2
29. 29.What is the smallest number you must add to 236 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
30. 30.What is the remainder when 314 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 2
31. 31.What is the smallest number you must add to 266 to make it divisible by 9? (a) 5 (b) 3 (c) 6 (d) 4
32. 32.What is the smallest number you must add to 671 to make it divisible by 12? (a) 3 (b) 1 (c) 2 (d) 11
33. 33.Which of these numbers is divisible by 9? (a) 6223 (b) 6234 (c) 6216 (d) 6219
34. 34.Which of these numbers is divisible by 4? (a) 2107 (b) 2111 (c) 2113 (d) 2108
35. 35.What is the units digit of 8^20? (a) 6 (b) 8 (c) 4 (d) 2
36. 36.What is the smallest number you must add to 50 to make it divisible by 3? (a) 2 (b) 1
37. 37.What is the smallest number you must add to 153 to make it divisible by 6? (a) 5 (b) 4 (c) 2 (d) 3
38. 38.What is the units digit of 7^33? (a) 7 (b) 3 (c) 1 (d) 9
39. 39.What is the smallest number you must add to 53 to make it divisible by 3? (a) 1 (b) 2
40. 40.Which of these numbers is divisible by 4? (a) 2295 (b) 2287 (c) 2291 (d) 2288

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