Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 838 · 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 7^35? (a) 1 (b) 9 (c) 3 (d) 7
2. 2.What is the remainder when 399 is divided by 8? (a) 7 (b) 5 (c) 6 (d) 1
3. 3.Which of these numbers is divisible by 3? (a) 1183 (b) 1186 (c) 1181 (d) 1182
4. 4.What is the smallest number you must add to 867 to make it divisible by 12? (a) 3 (b) 10 (c) 8 (d) 9
5. 5.What is the units digit of 7^21? (a) 9 (b) 7 (c) 3 (d) 1
6. 6.What is the units digit of 4^32? (a) 6 (b) 5 (c) 7 (d) 4
7. 7.What is the units digit of 8^12? (a) 6 (b) 4 (c) 2 (d) 8
8. 8.Which of these numbers is divisible by 9? (a) 1305 (b) 1310 (c) 1299 (d) 1318
9. 9.Which of these numbers is divisible by 4? (a) 2241 (b) 2238 (c) 2245 (d) 2240
10. 10.What is the units digit of 8^11? (a) 6 (b) 8 (c) 4 (d) 2
11. 11.What is the remainder when 241 is divided by 10? (a) 9 (b) 2 (c) 0 (d) 1
12. 12.What is the units digit of 9^41? (a) 1 (b) 7 (c) 9 (d) 8
13. 13.Which of these numbers is divisible by 9? (a) 6479 (b) 6495 (c) 6480 (d) 6488
14. 14.What is the units digit of 2^12? (a) 2 (b) 8 (c) 4 (d) 6
15. 15.What is the smallest number you must add to 385 to make it divisible by 6? (a) 4 (b) 5 (c) 1 (d) 3
16. 16.Which of these numbers is divisible by 3? (a) 2093 (b) 2098 (c) 2094 (d) 2095
17. 17.What is the smallest number you must add to 503 to make it divisible by 12? (a) 11 (b) 2 (c) 3 (d) 1
18. 18.What is the remainder when 215 is divided by 11? (a) 7 (b) 6 (c) 8 (d) 5
19. 19.Which of these numbers is divisible by 9? (a) 3774 (b) 3785 (c) 3771 (d) 3770
20. 20.Which of these numbers is divisible by 3? (a) 1622 (b) 1618 (c) 1624 (d) 1620
21. 21.What is the smallest number you must add to 224 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 2
22. 22.What is the smallest number you must add to 248 to make it divisible by 12? (a) 4 (b) 8 (c) 5 (d) 3
23. 23.Which of these numbers is divisible by 9? (a) 1941 (b) 1959 (c) 1944 (d) 1946
24. 24.What is the smallest number you must add to 78 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
25. 25.Which of these numbers is divisible by 6? (a) 3733 (b) 3741 (c) 3732 (d) 3731
26. 26.What is the units digit of 9^15? (a) 9 (b) 7 (c) 1 (d) 8
27. 27.What is the remainder when 480 is divided by 9? (a) 6 (b) 2 (c) 4 (d) 3
28. 28.What is the smallest number you must add to 616 to make it divisible by 12? (a) 4 (b) 7 (c) 9 (d) 8
29. 29.What is the remainder when 658 is divided by 9? (a) 0 (b) 2 (c) 1 (d) 8
30. 30.What is the units digit of 8^39? (a) 2 (b) 4 (c) 8 (d) 6
31. 31.What is the units digit of 3^27? (a) 9 (b) 7 (c) 1 (d) 3
32. 32.What is the smallest number you must add to 550 to make it divisible by 12? (a) 3 (b) 10 (c) 2 (d) 1
33. 33.What is the smallest number you must add to 307 to make it divisible by 4? (a) 2 (b) 3 (c) 1
34. 34.What is the units digit of 8^48? (a) 2 (b) 4 (c) 8 (d) 6
35. 35.What is the units digit of 8^16? (a) 8 (b) 2 (c) 6 (d) 4
36. 36.Which of these numbers is divisible by 3? (a) 1969 (b) 1965 (c) 1966 (d) 1963
37. 37.What is the smallest number you must add to 501 to make it divisible by 7? (a) 2 (b) 4 (c) 3 (d) 5
38. 38.Which of these numbers is divisible by 3? (a) 422 (b) 418 (c) 416 (d) 417
39. 39.What is the units digit of 3^15? (a) 1 (b) 7 (c) 3 (d) 9
40. 40.Which of these numbers is divisible by 11? (a) 1829 (b) 1838 (c) 1818 (d) 1826

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