Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 850 · 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 3^31? (a) 3 (b) 9 (c) 1 (d) 7
2. 2.What is the smallest number you must add to 345 to make it divisible by 9? (a) 5 (b) 6 (c) 3 (d) 7
3. 3.What is the units digit of 2^31? (a) 2 (b) 8 (c) 4 (d) 6
4. 4.Which of these numbers is divisible by 11? (a) 3157 (b) 3169 (c) 3149 (d) 3159
5. 5.Which of these numbers is divisible by 11? (a) 7477 (b) 7500 (c) 7481 (d) 7480
6. 6.What is the smallest number you must add to 229 to make it divisible by 11? (a) 3 (b) 2 (c) 1 (d) 9
7. 7.What is the smallest number you must add to 623 to make it divisible by 9? (a) 7 (b) 8 (c) 2 (d) 6
8. 8.What is the remainder when 208 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 5
9. 9.What is the smallest number you must add to 835 to make it divisible by 11? (a) 1 (b) 2 (c) 3 (d) 10
10. 10.What is the units digit of 2^56? (a) 6 (b) 2 (c) 4 (d) 8
11. 11.What is the units digit of 4^46? (a) 7 (b) 4 (c) 5 (d) 6
12. 12.Which of these numbers is divisible by 6? (a) 2154 (b) 2153 (c) 2156 (d) 2163
13. 13.What is the smallest number you must add to 131 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
14. 14.What is the units digit of 4^50? (a) 4 (b) 7 (c) 6 (d) 5
15. 15.What is the remainder when 109 is divided by 7? (a) 3 (b) 6 (c) 4 (d) 5
16. 16.What is the remainder when 223 is divided by 3? (a) 2 (b) 0 (c) 1
17. 17.What is the remainder when 336 is divided by 9? (a) 2 (b) 3 (c) 6 (d) 4
18. 18.Which of these numbers is divisible by 4? (a) 1069 (b) 1072 (c) 1079 (d) 1073
19. 19.Which of these numbers is divisible by 11? (a) 5003 (b) 5005 (c) 5006 (d) 5026
20. 20.Which of these numbers is divisible by 3? (a) 1822 (b) 1819 (c) 1817 (d) 1818
21. 21.Which of these numbers is divisible by 11? (a) 9143 (b) 9153 (c) 9135 (d) 9141
22. 22.What is the remainder when 201 is divided by 5? (a) 0 (b) 1 (c) 4 (d) 2
23. 23.Which of these numbers is divisible by 6? (a) 3702 (b) 3698 (c) 3712 (d) 3703
24. 24.Which of these numbers is divisible by 3? (a) 578 (b) 571 (c) 573 (d) 575
25. 25.Which of these numbers is divisible by 6? (a) 4384 (b) 4387 (c) 4380 (d) 4378
26. 26.What is the smallest number you must add to 199 to make it divisible by 3? (a) 1 (b) 2
27. 27.What is the units digit of 7^25? (a) 3 (b) 1 (c) 7 (d) 9
28. 28.Which of these numbers is divisible by 3? (a) 2690 (b) 2688 (c) 2693 (d) 2687
29. 29.What is the units digit of 4^45? (a) 6 (b) 3 (c) 4 (d) 5
30. 30.What is the units digit of 9^21? (a) 7 (b) 8 (c) 1 (d) 9
31. 31.What is the smallest number you must add to 88 to make it divisible by 3? (a) 2 (b) 1
32. 32.What is the remainder when 222 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
33. 33.What is the units digit of 8^20? (a) 4 (b) 8 (c) 2 (d) 6
34. 34.What is the smallest number you must add to 672 to make it divisible by 11? (a) 8 (b) 10 (c) 1 (d) 9
35. 35.What is the units digit of 3^47? (a) 7 (b) 1 (c) 9 (d) 3
36. 36.Which of these numbers is divisible by 11? (a) 2530 (b) 2524 (c) 2547 (d) 2533
37. 37.Which of these numbers is divisible by 9? (a) 2631 (b) 2642 (c) 2628 (d) 2626
38. 38.What is the units digit of 7^19? (a) 3 (b) 9 (c) 1 (d) 7
39. 39.Which of these numbers is divisible by 9? (a) 7868 (b) 7860 (c) 7866 (d) 7878
40. 40.What is the remainder when 151 is divided by 8? (a) 5 (b) 6 (c) 1 (d) 7

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