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## What number am I? – Prime numbers and perfect squares

Math Teaser

I am a perfect square. My square root is a prime number, less than 100, and the digits of the square root add up to a perfect square. My own digits add up to a prime number.

To solve this, we need to first determine what the options for the square root of the number. We know that it is a prime number and less than 100, so that is our starting list:

##### List of prime numbers 1-100

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97

Look through the list and determine which of these have digits that add up to a perfect square. A perfect square is an integer that has a square root that is an integer, or it is the product of an integer multiplied by itself.

13, 31 add up to 4 and 79, 97 add up to 16. These are the possible numbers for the square root of our number. We know our number is a perfect square, so we can square these numbers:

13 x 13 = 169
31 x 31 = 961
79 x 79 = 6241
97 x 97 = 9409

For each of the results, add up the digits. The mystery number is the only one whose digits add up to a prime number.

169 = 1 + 6 + 9 = 16
961 = 9 + 6 + 1 = 16
6241 = 6 + 2 + 4 + 1 = 13
9409 = 9 + 4 + 0 + 9 = 22

13 is the only resulting prime number, so the mystery number must be 6241