sum of its digits?

What is the smallest whole number that is equal to seven times the sum of its digits?


The answer is 21.

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1 solution

Matteo Damiano
Sep 29, 2018

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 a and b b be the digits of the number, which can be expressed as 10 a + b 10a+b .

10 a + b = 7 ( a + b ) 10a+b=7(a+b) , as stated in the question.

10 a + b = 7 a + 7 b 10a+b=7a+7b , by expanding the product.

10 a 7 a = 7 b b 10a-7a=7b-b , by rearranging the terms.

3 a = 6 b 3a=6b

a = 2 b a=2b

An integer solution is needed and the smallest is a = 2 a=2 and b = 1 b=1 .

The solution is 21 21 , and the conditions are verified.

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