Question 1161905: Suppose your textbook is opened at random and the product of the facing page numbers is 71556. To what page is the textbook opened?
Found 4 solutions by greenestamps, jim_thompson5910, ikleyn, MathTherapy:
Answer by greenestamps(13203)   (Show Source):
You can put this solution on YOUR website!
The two pages have consecutive page numbers.
We can find the page numbers using estimation and logical reasoning.
(1) Looking at the first three digits "715", we can see that they lie between 26^2=676 and 27^2 = 729. So the two page numbers are in the 260's.
(2) The units digit of the product is 6. The product of two consecutive integers has units digit 6 if the two units digits are either 2 and 3 or 7 and 8.
(3) Since 715 is closer to 729 than to 676, the two page numbers must be 627 and 628.
The answer is verified by multiplying those two numbers and seeing that the product is indeed 71556.
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Comments added after seeing three responses to your question made after I made mine....
In terms of efficiency, if you have a calculator, the quickest way to find the answer is as shown by tutor @ikleyn: take the square root of 71556 to find that it is about 627.5, making the two page numbers 627 and 628.
If you do not have a calculator, then estimation and logical reasoning will get you to the answer in far less time than the method shown by the other two tutors.
Finding the answer by finding the prime factorization of 71556 and then finding how to combine those factors to get two consecutive integers takes a lot of work. An average person will get the final answer to the problem in less time than it takes an average person to find the prime factorization -- not to mention the time it takes to play with that prime factorization to find the two page numbers.
Answer by jim_thompson5910(35256)  (Show Source):
You can put this solution on YOUR website!
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).
Answer by ikleyn(52816)  (Show Source):
You can put this solution on YOUR website! .
Take the square root of the number
 = 267.5.
The pages are 267 and 268. ANSWER
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By the way, you practically NEVER WILL FIND a book with such numeration of pages.
IT IS INCORRECT numeration.
Correct numeration has even number first and lesser (on the left facing page), than the second number (on the right facing page).
So the problem, as it is presented in the post, is a FAKE (!)
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See the lesson
- Page numbers on the left and right facing pages of an opened book
in this site, and learn the subject from there.
Happy learning (!)
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The moral is twofold for a student and for a "composer" of the Math problems.
For a student : Taking the square root is the quickest way to solve such problems.
For a composer : think what are you doing.
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Regarding the answer by @greenestamps, keep in mind and be aware, that his numbers are INCORRECT.
This problem IS DEFINITELY not for the mental solution, and the attempt by @greenestamps to solve it MENTALLY
and his failure CLEARLY PROVES it.
Answer by MathTherapy(10555)  (Show Source):
You can put this solution on YOUR website!
Suppose your textbook is opened at random and the product of the facing page numbers is 71556. To what page is the textbook opened?
Prime factors of 
Pages numbers: 
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