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I observed the sequence of perfect squares

$$1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,......$$

I noticed that either unit digit is 6 or tens place digit is even

Am I right or this doesn't follow for some number

I am trying to find such number or proving it right so please help me!

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    The down-vote is completely unnecessary!!! This is a good question! – 2017-01-09

1 Answers 1

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All numbers can be seen as $10a +b$, where $a\ge 0$ and $0\le b\le 9$. Its square writes $100a^2 + 20 a b + b^2$. The first term has no influence on the last two digits, so you need to consider only $20 a b + b^2$.

Look what happens at the tens place: the term $20 a b$ will necessarily give you an even number, and $b^2$ will give you something. The list of squares of numbers $0$ through $9$ ($1,4,9,16,25,36,49,64,81$) assures you that odd number at tens place comes with $6$ at units place; the final remark is that the term $20ab$ has no impact on the units digit.