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The distinct positive integers a, b, c, d are such that

* The product abcd is 40320

* ab + a + b = 322

* bc + b + c = 398

 

Find a, b, c, and d.

 Jun 2, 2020
 #1
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ab + a + b  =  322   --->   ab + b + a  =  322   --->   b(a + 1) + a  =  322   --->   b  =  (322 - a)/(a + 1)

                               --->  if a = 16, b = 18;  if a = 18, b = 16

 

bc + b + c  =  398   --->   b(c + 1) + c  =  398   --->   b  =  (398 - c)/(c + 1)

                               --->   if c = 6, b = 56;  if c = 56, b = 6;  if c = 18, b = 20;  if c = 20, b = 18

 

Since b can be 18 in both cases, let a = 16, b = 18, and c = 20   --->  which means that d = 7

 

I got these pairs of values by placing  y1 = (322 - x)/(x + 1)  and  y2 = (398 - x)/(x + 1)

into my calculator and scrolling down the table of values.

 Jun 2, 2020

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