# Determinate the integers x for which the fraction (3x+2)/(4x+5) can be simplified with 7.

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If the fraction can be simplified by 7, taht means that the numerator and denominator are multiple of 7:

3x + 2 = 7n

We'll isolate x to the left side and we'll keep in mind that we'll ahve to get an integer value for x:

3x = 7n - 2

x = (7n-2)/3

For x to be integer, than 7n - 2 has to be divided by 3:

If n = 2 => 7*2 - 2 = 14 - 2 = 12

12 is divided by 3 => x = 4 for n = 2

If n = 5 => x = 11

If n = 8 => x = 18...

Also, the denominator has to be multiple of 7:

4x + 5 = 7k => x = (7k-5)/4

We'll put x = 4 => 16 = 7k - 5 => 7k = 21 => k = 3

We'll put x = 11 => 44 = 7k-5 => 49 = 7k => k = 7

We'll put x = 18 => 72= 7k-5 => 77 = 7k => k = 11

**Therefore, the values of x :{4 ; 11 ; 18 ....} are the terms of an arithmetical progression, whose common difference is d = 7 and the first term is 4.**

Given p = (3x+2)/(4x+5) could be simplified by an integer 7.

So 3x+2 and 4x+5 should have a common factor 7.

Therefore 4x+5-(3x+2) = x+3 could be equal to 7 or its multiple.

So x = 7-3. So = 4 may be a solution.

Verification: 3*4+2 = 14 and and 4x+5= 4*4+4 = 21. So x= 4 is a solution.

So p = 14/21 could be simplified by 7 as 7 divides both numerator and denominator.

If 4 is a solution , then 4+7*n is also a solution, as the numerator 3x+2 = 3(4+7n)+2 =7(n+2) is a multiple of 7. And the denominator 4x+5 = (4+7n)+5 = 7(n+3) is a multiple of 7.

**Therefore the whole numbers that satisfy the given condtion are x = 4+7n . n = 0, 1,2,3.....**