Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 800 · 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 2^13? (a) 6 (b) 4 (c) 2 (d) 8
2. 2.What is the units digit of 2^42? (a) 8 (b) 2 (c) 6 (d) 4
3. 3.What is the remainder when 496 is divided by 12? (a) 8 (b) 4 (c) 3 (d) 5
4. 4.What is the remainder when 119 is divided by 5? (a) 3 (b) 4 (c) 2 (d) 1
5. 5.Which of these numbers is divisible by 6? (a) 3622 (b) 3624 (c) 3625 (d) 3631
6. 6.Which of these numbers is divisible by 3? (a) 1684 (b) 1681 (c) 1680 (d) 1679
7. 7.What is the smallest number you must add to 651 to make it divisible by 12? (a) 9 (b) 8 (c) 3 (d) 10
8. 8.What is the remainder when 682 is divided by 12? (a) 11 (b) 2 (c) 10 (d) 9
9. 9.What is the smallest number you must add to 86 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
10. 10.What is the remainder when 234 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
11. 11.What is the units digit of 3^40? (a) 3 (b) 1 (c) 9 (d) 7
12. 12.What is the units digit of 9^17? (a) 7 (b) 1 (c) 9 (d) 8
13. 13.What is the units digit of 7^41? (a) 9 (b) 7 (c) 3 (d) 1
14. 14.Which of these numbers is divisible by 9? (a) 6894 (b) 6911 (c) 6896 (d) 6889
15. 15.Which of these numbers is divisible by 9? (a) 7604 (b) 7586 (c) 7587 (d) 7588
16. 16.What is the units digit of 3^12? (a) 9 (b) 7 (c) 3 (d) 1
17. 17.What is the smallest number you must add to 276 to make it divisible by 9? (a) 4 (b) 2 (c) 6 (d) 3
18. 18.Which of these numbers is divisible by 6? (a) 1210 (b) 1201 (c) 1217 (d) 1206
19. 19.Which of these numbers is divisible by 11? (a) 4030 (b) 4023 (c) 4011 (d) 4015
20. 20.Which of these numbers is divisible by 6? (a) 958 (b) 963 (c) 952 (d) 954
21. 21.What is the remainder when 743 is divided by 10? (a) 4 (b) 3 (c) 7 (d) 2
22. 22.What is the smallest number you must add to 239 to make it divisible by 8? (a) 1 (b) 3 (c) 2 (d) 7
23. 23.What is the remainder when 143 is divided by 3? (a) 1 (b) 0 (c) 2
24. 24.What is the units digit of 2^15? (a) 2 (b) 6 (c) 8 (d) 4
25. 25.Which of these numbers is divisible by 9? (a) 4528 (b) 4548 (c) 4536 (d) 4538
26. 26.What is the remainder when 499 is divided by 8? (a) 4 (b) 5 (c) 3 (d) 2
27. 27.What is the smallest number you must add to 295 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 1
28. 28.Which of these numbers is divisible by 11? (a) 6195 (b) 6182 (c) 6183 (d) 6173
29. 29.What is the remainder when 579 is divided by 9? (a) 6 (b) 4 (c) 2 (d) 3
30. 30.Which of these numbers is divisible by 9? (a) 5012 (b) 4998 (c) 5004 (d) 5018
31. 31.What is the units digit of 9^19? (a) 1 (b) 7 (c) 8 (d) 9
32. 32.What is the remainder when 41 is divided by 3? (a) 0 (b) 2 (c) 1
33. 33.What is the smallest number you must add to 290 to make it divisible by 7? (a) 5 (b) 4 (c) 6 (d) 3
34. 34.Which of these numbers is divisible by 8? (a) 5624 (b) 5625 (c) 5633 (d) 5622
35. 35.What is the units digit of 2^26? (a) 2 (b) 6 (c) 4 (d) 8
36. 36.What is the smallest number you must add to 417 to make it divisible by 6? (a) 4 (b) 2 (c) 5 (d) 3
37. 37.What is the remainder when 152 is divided by 9? (a) 8 (b) 1 (c) 7 (d) 6
38. 38.Which of these numbers is divisible by 9? (a) 4660 (b) 4663 (c) 4653 (d) 4645
39. 39.What is the units digit of 4^40? (a) 5 (b) 6 (c) 7 (d) 4
40. 40.What is the units digit of 2^12? (a) 4 (b) 6 (c) 2 (d) 8

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