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 #1
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What is the sum of all integer values of x such that 67/(2x – 23) is an integer? 

 

The only values of x that satisfy the requirement are 45 and 12, so the sum of all values is 57

 

When x is 45, the denominator becomes 67 so the expression is 67/67 which is 1, an integer. 

When x is 12, the denominator becomes   1 so the expression is 67/1 which is 67, an integer.

 

Any number larger than 45 makes the denominator larger than 67 so the result is a fraction. 

 

Since 67 is a prime number, no denominator will divide into it an even number of times except 1 and 67. 

 

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 Mar 14, 2020
 #4
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Darn it, I forgot that there would also be two negative denominators.  My answer is incorrect.  See Nerd Wizard's answer below. 

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Guest Mar 15, 2020
 #2
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Since 67 is a prime number, there are only 2 integer results we can have: 1 and 67.

 

To get 67, the denominator should be 1.

2x - 23 = 1 

2x = 24

x=12

 

To get 1 as the integer result, the denominator should be 67.

2x - 23 = 67

2x = 90

x = 45

 

12 + 45 = 57

 Mar 14, 2020

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