Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 923 · 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 173 to make it divisible by 10? (a) 7 (b) 6 (c) 3 (d) 8
2. 2.What is the remainder when 318 is divided by 9? (a) 4 (b) 6 (c) 3 (d) 2
3. 3.What is the remainder when 182 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
4. 4.What is the smallest number you must add to 278 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
5. 5.What is the smallest number you must add to 217 to make it divisible by 11? (a) 3 (b) 2 (c) 8 (d) 4
6. 6.What is the smallest number you must add to 224 to make it divisible by 9? (a) 2 (b) 1 (c) 8 (d) 3
7. 7.What is the units digit of 8^53? (a) 8 (b) 4 (c) 6 (d) 2
8. 8.Which of these numbers is divisible by 8? (a) 2259 (b) 2248 (c) 2246 (d) 2251
9. 9.Which of these numbers is divisible by 4? (a) 3197 (b) 3202 (c) 3200 (d) 3205
10. 10.What is the remainder when 46 is divided by 3? (a) 2 (b) 1 (c) 0
11. 11.What is the units digit of 3^40? (a) 3 (b) 9 (c) 1 (d) 7
12. 12.What is the remainder when 181 is divided by 3? (a) 1 (b) 2 (c) 0
13. 13.Which of these numbers is divisible by 3? (a) 562 (b) 568 (c) 565 (d) 564
14. 14.What is the remainder when 102 is divided by 9? (a) 2 (b) 3 (c) 6 (d) 4
15. 15.What is the remainder when 106 is divided by 5? (a) 4 (b) 2 (c) 0 (d) 1
16. 16.What is the units digit of 9^29? (a) 8 (b) 7 (c) 9 (d) 1
17. 17.What is the remainder when 105 is divided by 9? (a) 7 (b) 6 (c) 5 (d) 3
18. 18.What is the smallest number you must add to 509 to make it divisible by 11? (a) 3 (b) 7 (c) 9 (d) 8
19. 19.What is the smallest number you must add to 91 to make it divisible by 3? (a) 2 (b) 1
20. 20.What is the remainder when 372 is divided by 9? (a) 4 (b) 2 (c) 3 (d) 6
21. 21.Which of these numbers is divisible by 6? (a) 1629 (b) 1620 (c) 1617 (d) 1625
22. 22.What is the smallest number you must add to 145 to make it divisible by 6? (a) 4 (b) 5 (c) 1 (d) 3
23. 23.What is the remainder when 162 is divided by 7? (a) 1 (b) 0 (c) 2 (d) 6
24. 24.Which of these numbers is divisible by 3? (a) 1187 (b) 1184 (c) 1190 (d) 1185
25. 25.What is the remainder when 402 is divided by 9? (a) 5 (b) 7 (c) 6 (d) 3
26. 26.What is the units digit of 2^12? (a) 2 (b) 4 (c) 8 (d) 6
27. 27.Which of these numbers is divisible by 4? (a) 1775 (b) 1772 (c) 1779 (d) 1771
28. 28.Which of these numbers is divisible by 11? (a) 9390 (b) 9412 (c) 9402 (d) 9394
29. 29.What is the smallest number you must add to 459 to make it divisible by 11? (a) 3 (b) 2 (c) 4 (d) 8
30. 30.What is the units digit of 8^14? (a) 8 (b) 4 (c) 2 (d) 6
31. 31.Which of these numbers is divisible by 11? (a) 5231 (b) 5219 (c) 5225 (d) 5238
32. 32.What is the units digit of 9^41? (a) 1 (b) 7 (c) 9 (d) 8
33. 33.What is the units digit of 9^19? (a) 1 (b) 7 (c) 8 (d) 9
34. 34.Which of these numbers is divisible by 6? (a) 1831 (b) 1846 (c) 1838 (d) 1836
35. 35.What is the smallest number you must add to 192 to make it divisible by 10? (a) 2 (b) 8 (c) 7 (d) 9
36. 36.Which of these numbers is divisible by 3? (a) 1252 (b) 1259 (c) 1254 (d) 1255
37. 37.Which of these numbers is divisible by 6? (a) 649 (b) 642 (c) 647 (d) 641
38. 38.Which of these numbers is divisible by 6? (a) 2930 (b) 2923 (c) 2928 (d) 2936
39. 39.What is the remainder when 234 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
40. 40.What is the smallest number you must add to 313 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1

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