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xoF = (nx)oC (Posted on 2012-12-31) Difficulty: 3 of 5
If x degrees Fahrenheit = nx degrees Celsius, where each of n and x is a nonzero integer, determine all values of n and x for which this is possible.

Keeping all the other conditions unaltered, how about:
x degrees Celsius = nx degrees Fahrenheit?

Note:
Fahrenheit to Celsius:
oC= 5*(oF - 32)/9

Celsius to Fahrenheit :
oF= (C*9/5) + 32

No Solution Yet Submitted by K Sengupta    
Rating: 3.0000 (2 votes)

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Solution Baby, it's cold and warm outside (spoiler) Comment 7 of 7 |
Here is the full solution, given that negative temperatures are allowed.

Rearrange the conversion formula, getting 160 = 5F - 9C

Part 1)  F = x, c = nx
            160 = 5x - 9nx
            x = 160/(5 - 9n)

            The divisors of 160 are
                   1, 2, 4, 8, 16 and 32,
            and 5, 10, 20, 40, 80, 160 
            and -1, -2, -4, -8, -16, -32,
            and -5, -10, -20, -40, -80, -160
            
            We are looking for divisors which equal 5 mod 9, 
            and the only ones are 32, 5, -4,  and -40

            5 - 9n = 32 gives n = -3, x = 5 = F,  C = -15
            5 - 9n = 5   gives n = 0, x = 32 = F,  C = 0  
            5 - 9n = -4  gives n = 1, x = -40 = F,  C = -40  
            5 - 9n = -40 gives n = 5, x = -4 = F, C = -20              


Part 1)  F = nx, c = x
            160 = 5nx - 9x
            x = 160/(5n - 9)
            
            5n - 9 is not a multiple of 5.
            The only divisors of 160 that are not a multiple of 5 are
                      1, 2, 4, 8, 16 and 32, 
               and -1, -2, -4, -8, -16, -32
            5n - 9 = 1 (mod 5), 
               and the only divisors that equal 1 mod 5 
               are 1 and 16 and -4

            5n - 9 = 1 gives n = 2, x = 160 = C, F = 320
            5n - 9 = 16 gives n = 5, x = 10 = C,  F = 50 
            5n - 9 = -4  gives n = 1, x = -40 = C, F = -40

  Posted by Steve Herman on 2013-01-01 10:50:43
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