If (5x)2=y202 where x,y are integers, then what are x and y ?

1 Answer
Jun 3, 2015

Expand and equate the irrational parts to help simplify and find:

x=8 and y=33

Explanation:

Expand the left hand side:

(5x)2

=52(2×5×x)+x

=2510x+x

=(x+25)10x

In order to eliminate the irrational 202 term with x and y being integers, we must have:

10x=202

Divide both sides by 10 to get:

x=22

So

x=(22)2=8

y=x+25=8+25=33


Footnote

Why is it possible to equate the irrational parts like this?

Consider the set of all numbers of the form a+b2 where a and b are rational numbers.

These representations are unique:

Suppose a+b2=c+d2

Then ac=(db)2

So if db we would find 2=acdb which is a rational number, but 2 is irrational.

So we must have d=b and hence a=c.