Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 94 · 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 remainder when 287 is divided by 9? (a) 1 (b) 6 (c) 8 (d) 7
2. 2.What is the units digit of 4^16? (a) 5 (b) 4 (c) 6 (d) 7
3. 3.What is the smallest number you must add to 301 to make it divisible by 4? (a) 3 (b) 1 (c) 2
4. 4.What is the smallest number you must add to 334 to make it divisible by 7? (a) 1 (b) 5 (c) 2 (d) 3
5. 5.What is the smallest number you must add to 49 to make it divisible by 3? (a) 2 (b) 1
6. 6.What is the remainder when 346 is divided by 11? (a) 6 (b) 4 (c) 5 (d) 7
7. 7.What is the units digit of 7^19? (a) 9 (b) 1 (c) 7 (d) 3
8. 8.Which of these numbers is divisible by 3? (a) 1301 (b) 1299 (c) 1298 (d) 1304
9. 9.What is the remainder when 154 is divided by 3? (a) 2 (b) 1 (c) 0
10. 10.What is the units digit of 7^17? (a) 1 (b) 3 (c) 7 (d) 9
11. 11.What is the remainder when 52 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
12. 12.What is the smallest number you must add to 491 to make it divisible by 7? (a) 5 (b) 1 (c) 6 (d) 4
13. 13.What is the units digit of 9^34? (a) 0 (b) 9 (c) 1 (d) 2
14. 14.What is the remainder when 411 is divided by 12? (a) 3 (b) 2 (c) 9 (d) 4
15. 15.Which of these numbers is divisible by 6? (a) 3604 (b) 3606 (c) 3614 (d) 3607
16. 16.Which of these numbers is divisible by 8? (a) 2750 (b) 2761 (c) 2758 (d) 2752
17. 17.What is the remainder when 279 is divided by 11? (a) 3 (b) 7 (c) 5 (d) 4
18. 18.Which of these numbers is divisible by 11? (a) 5700 (b) 5686 (c) 5687 (d) 5689
19. 19.Which of these numbers is divisible by 3? (a) 1446 (b) 1448 (c) 1444 (d) 1450
20. 20.Which of these numbers is divisible by 6? (a) 1478 (b) 1486 (c) 1491 (d) 1482
21. 21.Which of these numbers is divisible by 6? (a) 4927 (b) 4933 (c) 4924 (d) 4926
22. 22.Which of these numbers is divisible by 4? (a) 956 (b) 955 (c) 962 (d) 957
23. 23.What is the remainder when 173 is divided by 3? (a) 1 (b) 2 (c) 0
24. 24.What is the units digit of 9^58? (a) 9 (b) 2 (c) 1 (d) 0
25. 25.What is the units digit of 4^20? (a) 6 (b) 7 (c) 5 (d) 4
26. 26.What is the units digit of 2^57? (a) 2 (b) 8 (c) 4 (d) 6
27. 27.What is the units digit of 3^19? (a) 3 (b) 1 (c) 7 (d) 9
28. 28.What is the smallest number you must add to 453 to make it divisible by 6? (a) 5 (b) 3 (c) 4 (d) 2
29. 29.What is the units digit of 2^60? (a) 2 (b) 6 (c) 4 (d) 8
30. 30.What is the remainder when 368 is divided by 11? (a) 4 (b) 5 (c) 6 (d) 7
31. 31.What is the smallest number you must add to 155 to make it divisible by 4? (a) 1 (b) 3 (c) 2
32. 32.What is the smallest number you must add to 169 to make it divisible by 10? (a) 9 (b) 3 (c) 1 (d) 2
33. 33.What is the remainder when 773 is divided by 10? (a) 4 (b) 7 (c) 2 (d) 3
34. 34.What is the units digit of 3^50? (a) 9 (b) 3 (c) 7 (d) 1
35. 35.What is the smallest number you must add to 296 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
36. 36.Which of these numbers is divisible by 8? (a) 3527 (b) 3531 (c) 3528 (d) 3539
37. 37.What is the units digit of 2^48? (a) 2 (b) 4 (c) 6 (d) 8
38. 38.What is the smallest number you must add to 171 to make it divisible by 12? (a) 8 (b) 10 (c) 9 (d) 3
39. 39.What is the remainder when 277 is divided by 10? (a) 7 (b) 8 (c) 6 (d) 3
40. 40.What is the remainder when 396 is divided by 8? (a) 5 (b) 4 (c) 6 (d) 3

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