A six digit perfect square is represented as
AABCAB where the same letter represents the same digit, but a different letter represents a different digit.
It is also known that both these relationships are simultaneously satisfied:
- The numbers represented by each of AAB and CAB is a perfect square.
- The numbers represented by each of AB and ABC is a perfect square.
Determine the digits represented by
A,
B, and
C.
Note: Computer program solutions are welcome, but a semi-analytic(calculator+ p&p) method is preferred.
Lets start with just AB is a perfect square. There are six two-digit perfect squares, so AB=16, 25, 36, 49, 64, or 81.
Then AAB can only be one of 116, 225, 336, 449, 664, or 881. Only one of these is a perfect square: 225=15^2.
So A=2 and B=5.
Then look at ABC is a perfect square: 25C is a perfect square. This is a perfect square only when C=6: 256=16^2.
Now we can verify the last two quantities: CAB=625=25^2 and AABCAB=225625=475^2.
Solution
