Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 589 · 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 smallest number you must add to 303 to make it divisible by 11? (a) 7 (b) 5 (c) 6 (d) 4
2. 2.What is the remainder when 447 is divided by 6? (a) 2 (b) 5 (c) 4 (d) 3
3. 3.What is the remainder when 182 is divided by 9? (a) 7 (b) 3 (c) 1 (d) 2
4. 4.What is the smallest number you must add to 188 to make it divisible by 3? (a) 2 (b) 1
5. 5.What is the units digit of 3^26? (a) 3 (b) 7 (c) 1 (d) 9
6. 6.What is the units digit of 7^45? (a) 1 (b) 3 (c) 9 (d) 7
7. 7.Which of these numbers is divisible by 3? (a) 1897 (b) 1895 (c) 1892 (d) 1893
8. 8.What is the smallest number you must add to 160 to make it divisible by 3? (a) 2 (b) 1
9. 9.What is the smallest number you must add to 358 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
10. 10.What is the smallest number you must add to 121 to make it divisible by 3? (a) 2 (b) 1
11. 11.What is the remainder when 118 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
12. 12.What is the remainder when 135 is divided by 12? (a) 4 (b) 3 (c) 9 (d) 2
13. 13.Which of these numbers is divisible by 8? (a) 2208 (b) 2221 (c) 2206 (d) 2210
14. 14.What is the units digit of 3^30? (a) 1 (b) 3 (c) 7 (d) 9
15. 15.What is the units digit of 9^23? (a) 9 (b) 1 (c) 8 (d) 7
16. 16.What is the smallest number you must add to 59 to make it divisible by 4? (a) 2 (b) 3 (c) 1
17. 17.What is the smallest number you must add to 148 to make it divisible by 3? (a) 1 (b) 2
18. 18.What is the remainder when 76 is divided by 3? (a) 1 (b) 2 (c) 0
19. 19.What is the smallest number you must add to 55 to make it divisible by 4? (a) 3 (b) 1 (c) 2
20. 20.What is the units digit of 9^48? (a) 9 (b) 0 (c) 2 (d) 1
21. 21.What is the units digit of 2^15? (a) 6 (b) 4 (c) 2 (d) 8
22. 22.What is the remainder when 263 is divided by 11? (a) 1 (b) 9 (c) 8 (d) 10
23. 23.What is the smallest number you must add to 283 to make it divisible by 11? (a) 8 (b) 4 (c) 2 (d) 3
24. 24.What is the units digit of 3^19? (a) 9 (b) 7 (c) 3 (d) 1
25. 25.What is the units digit of 2^29? (a) 8 (b) 2 (c) 4 (d) 6
26. 26.What is the units digit of 8^15? (a) 6 (b) 2 (c) 8 (d) 4
27. 27.What is the smallest number you must add to 693 to make it divisible by 10? (a) 8 (b) 6 (c) 7 (d) 3
28. 28.What is the remainder when 773 is divided by 11? (a) 3 (b) 2 (c) 4 (d) 8
29. 29.What is the units digit of 2^21? (a) 4 (b) 8 (c) 2 (d) 6
30. 30.What is the smallest number you must add to 486 to make it divisible by 7? (a) 6 (b) 3 (c) 4 (d) 5
31. 31.What is the remainder when 714 is divided by 12? (a) 8 (b) 6 (c) 7 (d) 5
32. 32.What is the smallest number you must add to 580 to make it divisible by 11? (a) 2 (b) 4 (c) 8 (d) 3
33. 33.What is the remainder when 95 is divided by 3? (a) 1 (b) 0 (c) 2
34. 34.What is the units digit of 9^57? (a) 7 (b) 9 (c) 1 (d) 8
35. 35.What is the units digit of 9^25? (a) 1 (b) 8 (c) 9 (d) 7
36. 36.What is the units digit of 4^31? (a) 3 (b) 4 (c) 6 (d) 5
37. 37.What is the smallest number you must add to 613 to make it divisible by 11? (a) 3 (b) 2 (c) 4 (d) 8
38. 38.What is the smallest number you must add to 178 to make it divisible by 3? (a) 1 (b) 2
39. 39.What is the units digit of 4^43? (a) 5 (b) 6 (c) 3 (d) 4
40. 40.Which of these numbers is divisible by 9? (a) 4980 (b) 4986 (c) 4996 (d) 4993

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