Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 21 · 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 649 to make it divisible by 9? (a) 7 (b) 8 (c) 1 (d) 6
2. 2.What is the units digit of 3^24? (a) 9 (b) 7 (c) 1 (d) 3
3. 3.What is the remainder when 368 is divided by 11? (a) 6 (b) 7 (c) 4 (d) 5
4. 4.What is the remainder when 145 is divided by 6? (a) 5 (b) 0 (c) 2 (d) 1
5. 5.What is the remainder when 438 is divided by 10? (a) 7 (b) 9 (c) 8 (d) 2
6. 6.What is the remainder when 94 is divided by 7? (a) 4 (b) 5 (c) 3 (d) 2
7. 7.What is the smallest number you must add to 365 to make it divisible by 6? (a) 2 (b) 5 (c) 1 (d) 3
8. 8.What is the remainder when 706 is divided by 11? (a) 3 (b) 1 (c) 9 (d) 2
9. 9.Which of these numbers is divisible by 6? (a) 639 (b) 650 (c) 643 (d) 642
10. 10.What is the smallest number you must add to 369 to make it divisible by 8? (a) 5 (b) 6 (c) 7 (d) 1
11. 11.Which of these numbers is divisible by 4? (a) 2347 (b) 2354 (c) 2348 (d) 2350
12. 12.What is the smallest number you must add to 215 to make it divisible by 9? (a) 3 (b) 2 (c) 8 (d) 1
13. 13.Which of these numbers is divisible by 3? (a) 2065 (b) 2067 (c) 2068 (d) 2072
14. 14.What is the remainder when 435 is divided by 10? (a) 6 (b) 4 (c) 7 (d) 5
15. 15.What is the smallest number you must add to 136 to make it divisible by 3? (a) 2 (b) 1
16. 16.What is the remainder when 677 is divided by 11? (a) 7 (b) 5 (c) 6 (d) 8
17. 17.Which of these numbers is divisible by 11? (a) 2164 (b) 2156 (c) 2176 (d) 2146
18. 18.What is the remainder when 133 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
19. 19.Which of these numbers is divisible by 9? (a) 7662 (b) 7651 (c) 7650 (d) 7648
20. 20.What is the units digit of 9^46? (a) 0 (b) 9 (c) 1 (d) 2
21. 21.What is the smallest number you must add to 76 to make it divisible by 3? (a) 2 (b) 1
22. 22.Which of these numbers is divisible by 4? (a) 2000 (b) 2002 (c) 1997 (d) 2006
23. 23.What is the units digit of 4^14? (a) 5 (b) 4 (c) 7 (d) 6
24. 24.What is the units digit of 4^26? (a) 4 (b) 7 (c) 5 (d) 6
25. 25.Which of these numbers is divisible by 8? (a) 5206 (b) 5191 (c) 5192 (d) 5199
26. 26.What is the remainder when 213 is divided by 6? (a) 4 (b) 5 (c) 3 (d) 2
27. 27.What is the units digit of 2^24? (a) 6 (b) 8 (c) 4 (d) 2
28. 28.Which of these numbers is divisible by 6? (a) 638 (b) 630 (c) 628 (d) 631
29. 29.What is the remainder when 97 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
30. 30.What is the smallest number you must add to 621 to make it divisible by 8? (a) 3 (b) 5 (c) 4 (d) 2
31. 31.Which of these numbers is divisible by 4? (a) 3481 (b) 3474 (c) 3477 (d) 3476
32. 32.What is the smallest number you must add to 366 to make it divisible by 9? (a) 3 (b) 2 (c) 4 (d) 6
33. 33.What is the remainder when 553 is divided by 12? (a) 1 (b) 2 (c) 0 (d) 11
34. 34.Which of these numbers is divisible by 11? (a) 8492 (b) 8496 (c) 8487 (d) 8504
35. 35.What is the smallest number you must add to 347 to make it divisible by 6? (a) 2 (b) 1 (c) 5 (d) 3
36. 36.What is the units digit of 7^46? (a) 3 (b) 1 (c) 9 (d) 7
37. 37.What is the remainder when 225 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
38. 38.What is the units digit of 8^35? (a) 6 (b) 2 (c) 8 (d) 4
39. 39.Which of these numbers is divisible by 11? (a) 7213 (b) 7222 (c) 7216 (d) 7231
40. 40.What is the units digit of 7^31? (a) 7 (b) 1 (c) 9 (d) 3

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