Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 362 · 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) 1116 (b) 1124 (c) 1117 (d) 1111
2. 2.What is the remainder when 373 is divided by 7? (a) 3 (b) 1 (c) 5 (d) 2
3. 3.What is the units digit of 3^28? (a) 7 (b) 9 (c) 3 (d) 1
4. 4.Which of these numbers is divisible by 3? (a) 1651 (b) 1648 (c) 1650 (d) 1655
5. 5.What is the units digit of 4^58? (a) 4 (b) 7 (c) 5 (d) 6
6. 6.What is the units digit of 8^45? (a) 6 (b) 8 (c) 4 (d) 2
7. 7.What is the units digit of 2^17? (a) 6 (b) 4 (c) 2 (d) 8
8. 8.What is the smallest number you must add to 911 to make it divisible by 12? (a) 11 (b) 2 (c) 1 (d) 3
9. 9.What is the smallest number you must add to 283 to make it divisible by 4? (a) 1 (b) 2 (c) 3
10. 10.Which of these numbers is divisible by 9? (a) 756 (b) 768 (c) 750 (d) 758
11. 11.What is the remainder when 166 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
12. 12.What is the remainder when 119 is divided by 10? (a) 9 (b) 8 (c) 1 (d) 7
13. 13.What is the smallest number you must add to 82 to make it divisible by 3? (a) 1 (b) 2
14. 14.Which of these numbers is divisible by 9? (a) 2100 (b) 2111 (c) 2119 (d) 2106
15. 15.Which of these numbers is divisible by 3? (a) 2652 (b) 2656 (c) 2650 (d) 2653
16. 16.What is the units digit of 7^39? (a) 3 (b) 7 (c) 1 (d) 9
17. 17.What is the smallest number you must add to 542 to make it divisible by 11? (a) 9 (b) 3 (c) 8 (d) 7
18. 18.What is the units digit of 8^56? (a) 4 (b) 8 (c) 6 (d) 2
19. 19.Which of these numbers is divisible by 9? (a) 2580 (b) 2584 (c) 2574 (d) 2567
20. 20.What is the remainder when 358 is divided by 7? (a) 0 (b) 1 (c) 6 (d) 2
21. 21.What is the units digit of 8^28? (a) 4 (b) 2 (c) 6 (d) 8
22. 22.What is the units digit of 2^33? (a) 8 (b) 6 (c) 2 (d) 4
23. 23.What is the units digit of 2^15? (a) 4 (b) 2 (c) 8 (d) 6
24. 24.What is the units digit of 2^18? (a) 6 (b) 4 (c) 8 (d) 2
25. 25.Which of these numbers is divisible by 11? (a) 9314 (b) 9317 (c) 9321 (d) 9331
26. 26.Which of these numbers is divisible by 6? (a) 656 (b) 648 (c) 647 (d) 653
27. 27.Which of these numbers is divisible by 8? (a) 6699 (b) 6711 (c) 6696 (d) 6692
28. 28.What is the remainder when 52 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
29. 29.What is the smallest number you must add to 562 to make it divisible by 7? (a) 2 (b) 4 (c) 6 (d) 5
30. 30.What is the units digit of 3^25? (a) 7 (b) 1 (c) 9 (d) 3
31. 31.What is the remainder when 423 is divided by 11? (a) 7 (b) 6 (c) 5 (d) 4
32. 32.What is the units digit of 3^39? (a) 7 (b) 3 (c) 9 (d) 1
33. 33.What is the remainder when 619 is divided by 10? (a) 8 (b) 7 (c) 1 (d) 9
34. 34.What is the smallest number you must add to 169 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
35. 35.What is the units digit of 2^39? (a) 8 (b) 2 (c) 6 (d) 4
36. 36.What is the units digit of 4^60? (a) 6 (b) 5 (c) 4 (d) 7
37. 37.What is the remainder when 500 is divided by 9? (a) 7 (b) 5 (c) 6 (d) 4
38. 38.What is the units digit of 2^26? (a) 8 (b) 6 (c) 2 (d) 4
39. 39.What is the remainder when 238 is divided by 3? (a) 0 (b) 1 (c) 2
40. 40.What is the units digit of 3^29? (a) 7 (b) 1 (c) 3 (d) 9

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