The page numbers of a novel were arranged with 2871 numbers. How many pages were there in this novel?

Answer 1

Let's say this novel has x pages.

Since the page number used 2871 numbers, the following page numbers can be listed:

2 3 4 2871

The page number of each page was composed of numbers, and each number appeared in the previous number of the page number, and the number of times each number appeared was not repeated. Therefore, the arrangement of the numbers on each page was as follows:

2 × 1 + 3 × 1 + 4 × 1 + + 2870 × 1 + 2871 × 1 = 2871 × (2 + 1 + 4 + + 1) = 2871 × n

where n is the number of times the number appears in the page number.

Therefore, this novel has a total of x pages. According to the above calculation, we can get:

x = 2871 × n

Substituting x into the above formula gives:

x = 2871 × n × (2 + 1 + 4 + + 1) / 2

In the above formula, n × (2 + 1 + 4 + + 1) is the sum of the number of times the number appears in the page number divided by 2 is the average number of times the number appears in the page number.

The above formula was simplified to:

x = 2871 × n × (n + 1) / 2

Since n is an integral number, x must be a multiple of 2871. At the same time, because each number in the page number does not repeat, n must be an odd number.

Therefore, this novel had a total of 2871 pages.