Question 1161905
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An algebraic approach


x = unknown page number
x+1 = page number after page x


x*(x+1) = x^2+x = product of page numbers
x^2+x = 71556
x^2+x-71556 = 0


Use the quadratic formula (with a = 1, b = 1, c = -71556) to find the two solutions to be x = 267 and x = -268. Ignore the negative x value, because we can't have negative page numbers.


Therefore, the only practical solution is x = 267. The pages we've opened the book to are 267 and 268.
We see that 267*268 = 71556 which confirms the answer.


Factoring or graphing is an alternative to solve x^2+x-71556 = 0; though both aren't really feasible by hand. 


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A non-algebraic approach is to look at the prime factorization of 71556
71556 = 2*2*3*67*89


Then group those factors into two groups, call them left and right group
One such arrangement is
left group = {2,2,3}
right group = {67,89}
The ordering of any particular single grouping does not matter.


Then multiply out each of the two groups
left group's product = 2*2*3 = 12
right group's product = 67*89 = 5963
the results of each group will then be the two page numbers; however, notice that 12 and 5963 are not adjacent pages. Also, it is unlikely a textbook has 5963 pages or more.


Then another arrangement is
left group = {2,67,3}
right group = {2*89}
left group's product = first page number = 2*67*3 = 402
right group's product = second page number = 2*89 = 178
We again get non-adjacent page numbers. At least the values are reasonable for any textbook.


So the idea is to guess and check. It turns out after a bit of trial and error you should find that
left group = 3*89 = 267
right group = 2*2*67 = 268
I used the previous section to help get to this quickly. If you didn't have that as a guide, then it would take longer. Though it's still a somewhat feasible route.
The recommendation I have is to make sure 67 and 89 are not in the same group. Otherwise their product will multiply to 67*89 = 5963 (or larger if other factors are involved).
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