Answer: 25
Work Shown
12^2 = 144
13^2 = 169
169 - 144 = 25
If n = 25, then
which is a natural number.
Also, if n = 25, then
is also a natural number.
n = 25 is the smallest such number so that each expression results in a natural number. This is because 13^2 = 169 is the next largest perfect square compared to 12^2 = 144. Any other larger perfect squares will require larger values of n.
Other values of n that make each expression a natural number are:
n = 25, n = 145, n = 180, n = 340
There are likely infinitely many other such values of n.