Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 679 · 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 4^60? (a) 4 (b) 5 (c) 7 (d) 6
2. 2.What is the units digit of 7^52? (a) 3 (b) 9 (c) 1 (d) 7
3. 3.What is the units digit of 7^23? (a) 7 (b) 9 (c) 3 (d) 1
4. 4.What is the units digit of 3^27? (a) 3 (b) 1 (c) 9 (d) 7
5. 5.Which of these numbers is divisible by 9? (a) 7291 (b) 7281 (c) 7279 (d) 7286
6. 6.What is the smallest number you must add to 661 to make it divisible by 9? (a) 4 (b) 5 (c) 6 (d) 7
7. 7.Which of these numbers is divisible by 3? (a) 548 (b) 550 (c) 553 (d) 549
8. 8.What is the smallest number you must add to 186 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
9. 9.What is the units digit of 8^49? (a) 4 (b) 6 (c) 2 (d) 8
10. 10.Which of these numbers is divisible by 3? (a) 2402 (b) 2400 (c) 2405 (d) 2398
11. 11.What is the units digit of 3^50? (a) 9 (b) 3 (c) 1 (d) 7
12. 12.Which of these numbers is divisible by 3? (a) 190 (b) 193 (c) 196 (d) 192
13. 13.What is the remainder when 449 is divided by 6? (a) 1 (b) 4 (c) 5 (d) 3
14. 14.What is the smallest number you must add to 166 to make it divisible by 3? (a) 1 (b) 2
15. 15.Which of these numbers is divisible by 6? (a) 2162 (b) 2156 (c) 2160 (d) 2168
16. 16.What is the units digit of 7^58? (a) 9 (b) 1 (c) 3 (d) 7
17. 17.Which of these numbers is divisible by 4? (a) 3049 (b) 3048 (c) 3047 (d) 3055
18. 18.What is the smallest number you must add to 274 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
19. 19.What is the remainder when 281 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
20. 20.Which of these numbers is divisible by 9? (a) 2220 (b) 2230 (c) 2206 (d) 2214
21. 21.Which of these numbers is divisible by 9? (a) 1023 (b) 1006 (c) 1012 (d) 1008
22. 22.What is the remainder when 136 is divided by 3? (a) 0 (b) 1 (c) 2
23. 23.What is the remainder when 269 is divided by 11? (a) 6 (b) 7 (c) 5 (d) 4
24. 24.What is the units digit of 2^36? (a) 8 (b) 2 (c) 4 (d) 6
25. 25.What is the smallest number you must add to 896 to make it divisible by 12? (a) 3 (b) 4 (c) 5 (d) 8
26. 26.Which of these numbers is divisible by 3? (a) 1820 (b) 1825 (c) 1822 (d) 1821
27. 27.What is the smallest number you must add to 114 to make it divisible by 9? (a) 3 (b) 6 (c) 2 (d) 4
28. 28.What is the units digit of 3^42? (a) 7 (b) 9 (c) 3 (d) 1
29. 29.What is the remainder when 565 is divided by 12? (a) 1 (b) 2 (c) 11 (d) 0
30. 30.What is the smallest number you must add to 198 to make it divisible by 10? (a) 8 (b) 3 (c) 2 (d) 1
31. 31.Which of these numbers is divisible by 6? (a) 4026 (b) 4031 (c) 4036 (d) 4021
32. 32.What is the remainder when 790 is divided by 11? (a) 2 (b) 10 (c) 9 (d) 8
33. 33.What is the smallest number you must add to 67 to make it divisible by 3? (a) 2 (b) 1
34. 34.What is the smallest number you must add to 140 to make it divisible by 12? (a) 5 (b) 4 (c) 3 (d) 8
35. 35.What is the smallest number you must add to 253 to make it divisible by 9? (a) 8 (b) 7 (c) 1 (d) 6
36. 36.Which of these numbers is divisible by 6? (a) 4404 (b) 4405 (c) 4414 (d) 4400
37. 37.Which of these numbers is divisible by 8? (a) 2063 (b) 2074 (c) 2064 (d) 2065
38. 38.Which of these numbers is divisible by 3? (a) 1775 (b) 1773 (c) 1777 (d) 1771
39. 39.What is the smallest number you must add to 403 to make it divisible by 12? (a) 7 (b) 6 (c) 4 (d) 5
40. 40.What is the units digit of 9^55? (a) 7 (b) 9 (c) 8 (d) 1

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