Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 97 · 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.Which of these numbers is divisible by 6? (a) 874 (b) 879 (c) 870 (d) 866
2. 2.What is the smallest number you must add to 366 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
3. 3.What is the units digit of 2^36? (a) 4 (b) 8 (c) 2 (d) 6
4. 4.Which of these numbers is divisible by 4? (a) 1275 (b) 1279 (c) 1276 (d) 1283
5. 5.What is the units digit of 4^10? (a) 7 (b) 6 (c) 5 (d) 4
6. 6.What is the units digit of 8^32? (a) 8 (b) 4 (c) 2 (d) 6
7. 7.What is the remainder when 235 is divided by 10? (a) 5 (b) 4 (c) 6 (d) 7
8. 8.Which of these numbers is divisible by 6? (a) 4830 (b) 4838 (c) 4828 (d) 4833
9. 9.What is the smallest number you must add to 556 to make it divisible by 7? (a) 6 (b) 5 (c) 4 (d) 3
10. 10.What is the remainder when 423 is divided by 8? (a) 5 (b) 7 (c) 6 (d) 1
11. 11.What is the remainder when 309 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
12. 12.What is the units digit of 3^55? (a) 1 (b) 9 (c) 7 (d) 3
13. 13.What is the remainder when 452 is divided by 6? (a) 4 (b) 1 (c) 2 (d) 3
14. 14.What is the units digit of 2^39? (a) 2 (b) 6 (c) 8 (d) 4
15. 15.What is the units digit of 2^41? (a) 4 (b) 8 (c) 2 (d) 6
16. 16.Which of these numbers is divisible by 8? (a) 6693 (b) 6685 (c) 6702 (d) 6688
17. 17.What is the smallest number you must add to 141 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 2
18. 18.Which of these numbers is divisible by 6? (a) 5067 (b) 5060 (c) 5058 (d) 5056
19. 19.Which of these numbers is divisible by 3? (a) 163 (b) 160 (c) 162 (d) 166
20. 20.What is the remainder when 223 is divided by 9? (a) 2 (b) 6 (c) 7 (d) 8
21. 21.Which of these numbers is divisible by 3? (a) 2213 (b) 2209 (c) 2208 (d) 2206
22. 22.What is the units digit of 7^19? (a) 1 (b) 7 (c) 9 (d) 3
23. 23.What is the smallest number you must add to 55 to make it divisible by 3? (a) 1 (b) 2
24. 24.Which of these numbers is divisible by 8? (a) 2890 (b) 2885 (c) 2880 (d) 2878
25. 25.What is the remainder when 143 is divided by 8? (a) 7 (b) 6 (c) 1 (d) 5
26. 26.Which of these numbers is divisible by 6? (a) 3407 (b) 3408 (c) 3419 (d) 3411
27. 27.What is the smallest number you must add to 199 to make it divisible by 3? (a) 1 (b) 2
28. 28.What is the smallest number you must add to 801 to make it divisible by 10? (a) 8 (b) 7 (c) 1 (d) 9
29. 29.What is the units digit of 3^43? (a) 9 (b) 1 (c) 3 (d) 7
30. 30.What is the units digit of 2^60? (a) 6 (b) 4 (c) 2 (d) 8
31. 31.What is the remainder when 192 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
32. 32.Which of these numbers is divisible by 11? (a) 7580 (b) 7602 (c) 7590 (d) 7593
33. 33.What is the remainder when 382 is divided by 6? (a) 3 (b) 4 (c) 2 (d) 5
34. 34.What is the units digit of 8^19? (a) 8 (b) 2 (c) 6 (d) 4
35. 35.What is the smallest number you must add to 124 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
36. 36.What is the units digit of 2^52? (a) 8 (b) 2 (c) 4 (d) 6
37. 37.What is the smallest number you must add to 274 to make it divisible by 6? (a) 3 (b) 4 (c) 1 (d) 2
38. 38.What is the remainder when 467 is divided by 12? (a) 11 (b) 10 (c) 9 (d) 1
39. 39.Which of these numbers is divisible by 4? (a) 746 (b) 741 (c) 739 (d) 740
40. 40.What is the units digit of 4^13? (a) 4 (b) 5 (c) 3 (d) 6

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