What is the smallest whole number that is equal to seven times the sum of its digits?
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There can't be any one-digit solution, because every one-digit number is equal to the sum of its digits.
Let's explore the case for two-digits numbers.
Let the a and b be the digits of the number, which can be expressed as 1 0 a + b .
1 0 a + b = 7 ( a + b ) , as stated in the question.
1 0 a + b = 7 a + 7 b , by expanding the product.
1 0 a − 7 a = 7 b − b , by rearranging the terms.
3 a = 6 b
a = 2 b
An integer solution is needed and the smallest is a = 2 and b = 1 .
The solution is 2 1 , and the conditions are verified.