An ancienct manuscript is on display in a museum case. To what pages is the manusript open of the product of the page numbers is 9, 312?
n(n-1) = 9312
n^2 - n - 9312 = 0
n= (1/2)(1 +/- sqrt(1+37248))
n = 1/2 +/- 193/2
n = 194/2 or a negative number
n = 97
n-1 = 96
check
96*97 =
By the way, you could get there faster by guessing
sqrt (10,000) is 100
so it is a little under 100
try 95 and 96 --> 9120 too small
try 96 and 97 ---> 9312 score!
To find out the page numbers that the ancient manuscript is open to, we need to determine two numbers whose product is 9,312.
Step 1: Find the prime factorization of 9,312.
9,312 = 2 x 2 x 2 x 2 x 3 x 7 x 11
Step 2: Group the factors into pairs that multiply to give the product.
(2 x 2 x 3) x (2 x 2 x 7 x 11)
Step 3: Determine the possible page numbers.
The pairs are (12 x 308) and (44 x 84).
Therefore, the manuscript is open to the pages 12 and 308 or the pages 44 and 84.
To find the pages to which the ancient manuscript is open, we need to find two page numbers that, when multiplied, result in a product of 9,312.
First, let's break down the product 9,312 into its prime factors:
9,312 = 2 * 2 * 2 * 2 * 2 * 19 * 19
From this prime factorization, we can see that 9,312 can be represented as the multiplication of two numbers:
9,312 = (2 * 2 * 2 * 2 * 2) * (19 * 19)
Simplifying further, we have:
9,312 = 32 * 19 * 19
Now, we are looking for two page numbers whose product is equal to 9,312. Since the product involves the number 32, one of the page numbers must be a multiple of 32.
To find the possibilities, we can start by listing factors of 9,312 that are multiples of 32:
32 * 1 = 32
64 * 1 = 64
96 * 1 = 96
128 * 1 = 128
...
...
...
928 * 1 = 928
Now, let's check if any of these page numbers multiplied by another number gives a product of 9,312:
32 * 291 = 9,312 (so, page numbers could be 32 and 291)
64 * 146 = 9,312 (so, page numbers could be 64 and 146)
96 * 97 = 9,312 (so, page numbers could be 96 and 97)
128 * 73 = 9,312 (so, page numbers could be 128 and 73)
...
Based on this analysis, the manuscript could open to pages 32 and 291, 64 and 146, 96 and 97, or 128 and 73.
However, without further information or context, it is impossible to determine which specific page numbers the ancient manuscript is open to.