SOLUTION: A student opens a mathematics book to two facing pages. The product of the page numbers is 2162. Find the page numbers.

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Question 291299: A student opens a mathematics book to two facing pages. The product of the page numbers is 2162. Find the page numbers.
Found 2 solutions by checkley77, Grinnell:
Answer by checkley77(12844) About Me  (Show Source):
You can put this solution on YOUR website!
x(x+1)=2162 [short cut - find the square root of the integer then add & subtract approximately 0.5 for the 2 answers.]
x^2+x-2162=0
(x-46)(x+47)=0
x-46=0
x=46 the number of the left hand page.
46+1=47 the number of the right hand page.
Proof:
46*47=2162
2162=2162

Answer by Grinnell(63) About Me  (Show Source):
You can put this solution on YOUR website!
This IS a nasty problem. It can be solved. I just have to grind it out (WITHOUT A CALCULATOR)! Let's go!
I'm gonna go fast, so hold on!
we have x(x+1)=2162
x^2+x-2162=0
We know that a product with the difference of 1=2162.
Where do we start? The numbers are close, so how about trying to figure
out close squares? 21 x 100=2100 pretty close --10 x square root of 21.
square root of 21 is between 4 and 5. 10 x 4 is 40! square this, we get 1600.
What two numbers between 40 and 50 (2500 is too large) are consecutive whose products in the ones' column end in a 2? try 43 and 44---1892, too small 48 and 49---2352, too large
46 and 47---2162---BINGO
IMORTANT--I DON'T HAVE A CALCULATOR!!!! YOU CAN SHOOT THIS THRU THE QUAD AND YOU END UP WITH LOOKING FOR THE SQUARE ROOT OF 8649. WITH A CALCULATOR THIS IS EASY, BUT BY PLUGGING AWAY AND NARROWING DOWN THE PRODUCTS OF 2162 (FROM INITIALLY EXTRAPOLATING THE SQUARE ROOT ACTUALLY) WE GET THE ANSWER. THE WAY I DID THIS IS PROBABLY NOT THE WAY YOUR TEACHER WANTS YOU TO SOLVE IT. BUT GO OVER WHAT I DID. I ASKED MYSELF THE QUESTION, WHAT TWO NICE SQUARES OF NUMBERS DOES 2162 FALL BETWEEN--40 AND 50. AND, WHAT NUMBERS BETWEEN 40 AND 50 WHEN MULTIPLIED TOGETHER WILL YIELD A PRODUCT ENDING WITH THE NUMBER TWO--2 IN THE ONES' COLUMN