Find the sum
Number Theory
Level
4
Find the sum of all integers equal to the square of the sum of the digits of the number.
The answer is 82.
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Suppose such a number exists, and let that number be a 1 a 2 . . . a n . From the question, we have: a 1 a 2 . . . a n = ( a 1 + a 2 + . . . + a n ) 2 ⇒ L H S > 1 0 n − 1 , R H S < ( 1 0 n ) 2
In order for the equality to hold, the following must be true:
( 1 0 n ) 2 > 1 0 n − 1
It is evident that n can only be 1, 2 or 3. Hence, we have bounded the question sufficiently and we can check the square of every positive integer less than 28 to find the numbers satisfying the conditions.