What is the sum of all two-digit whole numbers whose squares end with the digits 21?

2 Answers
Jul 19, 2016

200

Explanation:

A square number ending in a '1' can only be produced by squaring a number ending in a '1' or a '9'. Source. This helps a lot in the search. Quick bit of number crunching gives:

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from our table we can see that

112=121

392=1521

612=3721

892=7921

So 11+39+61+89=200

Jul 19, 2016

200

Explanation:

If the last digits of a square of a two digit number are 21, unit's digit is either 1 or 9.

Now, if tens digit is a and units digit is 1, it is of type 100a2+20a+1 and we can have last two digits as 21 if a is 1 or 6 i.e. numbers are 10+1=11 and 60+1=61.

If ten's digit is b and unit digit is 9, it is of type 100b220b+1 and we can have last two digits as 21 if b is 4 or 9 i.e. numbers are 401=39 and 901=89.

Hence, sum of all such two digit numbers is

11+39+61+89=200