Since \(3^{\dfrac{1}{3}}>4^{\dfrac{1}{4}}>5^{\dfrac{1}{5}}>...\) we have \(y^z\ge z^y\) if \(y\ge3\).

Since \(1\le x\le y\), the only possible values for (x, y) are (1,1), (1,2), (2,2). These lead to the equations \(1+1=z,1+2^x=z,4+2^z=z^2\).

The firt equation has the unique solution (1,1,2).

The second equarion has no solution because \(2^x>z\).

The third equation has also no solution since \(2^x\ge z^2,z\ge4\) and (2,2,3) is not a solution.

So the equation has unique solution (1,1,2).

Since 313>414>515>... we have yz≥zy if y≥3

.

Since 1≤x≤y

, the only possible values for (x, y) are (1,1), (1,2), (2,2). These lead to the equations 1+1=z,1+2x=z,4+2z=z2

.

The firt equation has the unique solution (1,1,2).

The second equarion has no solution because 2x>z

.

The third equation has also no solution since 2x≥z2,z≥4

 and (2,2,3) is not a solution.

So the equation has unique solution (1,1,2).

Carter selected this answer.

chứng tỏ rằng số:A=54^5-54^4 chia hết cho 53