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Square Root and Sum of Digits

Date: 07/25/2008 at 16:06:27
From: Kenny
Subject: Is the number 81 the only number whose square root and sum a

Is the number 81 the only number whose square root and sum are the same?  

Ie. square root = 9
    sum   8 + 1 = 9



Date: 07/25/2008 at 19:09:25
From: Doctor Achilles
Subject: Re: Is the number 81 the only number whose square root and sum a

Hi Kenny,

Thanks for writing to Dr. Math.

There is only one other number that has that property.

You can show that there is no number between 10,000 and 1,000,000 that
has that property.

The sum of digits for numbers between 10,000 and 1,000,000 can be as
small as 1 or as large as 54 (as with 999,999).

The square roots of numbers between 10,000 and 1,000,000 can be as
small as 100 or as large as 1,000.

Because the smallest possible square root (100) is larger than the
largest possible sum of digits (54), there cannot be a number between
10,000 and 1,000,000 which has the property that square root = sum of
digits.

You can also show that no number between 1,000,000 and 100,000,000 has
that property.

If we look from 1,000,000 to 100,000,000 the range of square roots is
1,000 to 10,000 and the range of sums of digits is 1 to 72.  Again,
there is no overlap.

In fact, no number greater than 10,000 can have the property:

For any pair of numbers: A and B, where A = 10^2n and B = A*100, the
range of square roots between A and B is 10^n to 10^(n+1) and the
range of sums of digits is 1 to 9*(2n+2).  When n is greater than 1
then 9*(2n+2) must be less than 10^n.

Therefore, we only need to worry about numbers less than 10^(2*2),
i.e. less than 10,000.

Doctor Ali here at the Math Forum has tested with a computer and 
found that there is no number between 100 and 10,000 that has the
property that its square root equals the sum of its digits.

So we only need to check perfect squares that are less than 100.  You
have found that it works for 81.  That leaves us with only a few
numbers to test:

  64, 49, 36, 25, 16, 9, 4, and 1

Which of these has the property that the square root equals the sum of
digits?

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Number Theory

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