If x and y are positive integers and (5^x)−(5^y)=(2^y−1)∗(5^x−1), what is the value of xy?
If x and y are positive integers and (5^x)−(5^y)=(2^y−1)∗(5^x−1), what is the value of xy?
Answer/Solution
12
Steps/Work
x and y are positive integers which means we will have clean numbers. On the right hand side, you have a 2 as a factor while it is not there on the left hand side. Can a 2 be generated on the left hand side by the subtraction? Here I am thinking that if we take 5^y common on the left hand side, I might be able to get a 2.
56y(56^x−y −1)=2^y−1∗5^x−1
Now I want only 2s and 5s on the left hand side. If x-y is 1, then (5^x−y−1) becomes 4 which is 2^2. If instead x - y is 2 or more, I will get factors such as 3, 13 too. So let me try putting x - y = 1 to get
5^y(2^2)=2^y−1∗5^x−1
This gives me y - 1 = 2
y = 3
x = 4
Check to see that the equations is satisfied with these values. Hence xy = 12
Answer (E)
56y(56^x−y −1)=2^y−1∗5^x−1
Now I want only 2s and 5s on the left hand side. If x-y is 1, then (5^x−y−1) becomes 4 which is 2^2. If instead x - y is 2 or more, I will get factors such as 3, 13 too. So let me try putting x - y = 1 to get
5^y(2^2)=2^y−1∗5^x−1
This gives me y - 1 = 2
y = 3
x = 4
Check to see that the equations is satisfied with these values. Hence xy = 12
Answer (E)