SANSKAR'S OG PROBLEMS
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Problem 1
Let be a 2-digit positive integer satisfying
. Find
.
Solution 1 by ddk001 (Casework)
Case 1:
In this case, we have
.
If , we must have
, but this contradicts the original assumption of , so hence we must have
.
With this in mind, we consider the unit digit of .
Subcase 1.1:
In this case, we have that
.
There is no apparent contradiction here, so we leave this as it is.
Subcase 1.2:
In this case, we have that
.
This contradicts with the fact that , so this is impossible.
Subcase 1.3:
In this case, we have that
.
However, this is impossible for all .
Subcase 1.4:
In this case, we have that
.
Again, this yields , which, again, contradicts
.
Hence, we must have .
Problem2
For any positive integer ,
>1 can
be a perfect square? If yes, give one such
. If no, then prove it.