SOLUTION: In the equation : x^a - x^b = z , where 'x' & 'z' is a positive integer, where a>b and 'a' & 'b' are positive integers. Can we find the value of 'a' & 'b' ? when the value of 'z' a

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: In the equation : x^a - x^b = z , where 'x' & 'z' is a positive integer, where a>b and 'a' & 'b' are positive integers. Can we find the value of 'a' & 'b' ? when the value of 'z' a      Log On


   

Question 142348: In the equation : x^a - x^b = z , where 'x' & 'z' is a positive integer, where a>b and 'a' & 'b' are positive integers. Can we find the value of 'a' & 'b' ? when the value of 'z' and 'x' is given. Is there any method. For example given eqn 2^a - 2^b = 32512. Here can we find that a=15 and b=8.
For example given eqn 3^a - 3^b = 6318. Here can we find that a=8 and b=5.
I need method to find a and b. This is the question of finding two variables in an equation. Please help me to know this. If possible please find URL where solution is present.(FROM SHIVA)
Answer by scott8148(6628) About Me  (Show Source):
You can put this solution on YOUR website!
x^a-x^b=x^b[x^(a-b)-1] __ x^(a-b)-1=z/(x^b)

x^b should be the highest power of x that evenly divides into z

2^a - 2^b = 32512 __ 2^8(2^(15-8)-1)=32512 __ 2^7-1=127 __ 2^7=128

3^a - 3^b = 6318 __ 3^5(3^(8-5)-1)=6318 __ 3^3-1=26 __ 3^3=27