The only two digited perfect squares are 16,25,36,49,64 and 81.Since no square stats or begins with a 2 or 5,we remove 25 from the list.
A prime number always starts with an odd digit.Therefore the only option is 49 and 81.Also no square ends in 8.So the last two digit of the number required is 49.The only two digited square ending in 4 is 64. So the number becomes ....649.The only combination are ..3649 and ..1649.But there is no square terminating with 3.So the only left option is ..1649.Thus the answer becomes 81649 which certainly is a prime number.
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