The number of digits used to write all the page in a book is 35. How many pages has the book?
To solve this problem, let's break it down step by step.
First, let's assume that each page in the book has the same number of digits. Let's call this number of digits "x".
Next, let's determine how many pages we can write using one digit. Since we have 10 possible digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), we can represent 10 different numbers using one digit. Therefore, we can write 10 pages using one digit.
Now, if we use two digits, we can represent 100 different numbers (10 x 10). So, using two digits, we can write 100 pages.
By continuing this pattern, we can write:
- 10 pages using one digit
- 100 pages using two digits
- 1000 pages using three digits
- 10000 pages using four digits
- ...
Now, we need to find the number of pages using a total of 35 digits. We can do this by finding the highest number of pages we can write using a certain number of digits and then subtracting from the total.
Since the number of digits used is 35, we know that it cannot be more than 10000 (as 10000 requires 5 digits).
Therefore, the number of pages using four digits is the highest possible number of pages. Using four digits, we can write 10000 pages.
Now, let's subtract 10000 pages from the total:
35 - 10000 = -9965
Since we obtain a negative number, it is clear that using four digits is not possible. We need to consider using fewer digits.
Using three digits, we can represent 1000 pages. Subtracting 1000 from the total:
35 - 1000 = -965
Again, we obtain a negative number. This means that using three digits is not possible either.
Using two digits, we can represent 100 pages. Subtracting 100 from the total:
35 - 100 = -65
Once again, we obtain a negative number.
Lastly, using one digit, we can represent 10 pages. Subtracting 10 from the total:
35 - 10 = 25
Finally, we have a positive number. This means that using one digit is the correct scenario.
Therefore, the book must have 10 pages in total.