What is the only number that is exactly twice the sum of its digits? Solution below:

June 27, 2013 — Leave a comment

First, our number has digits, so it must be greater than 10.

Second, our number cannot be greater than 36 (since the maximum value of each digit is 9, and 2 x (9 + 9) = 36.

Third, our number must be even (since it’s been doubled.)

So, a search of our even numbers between 10 and 36 yields the solution 18.

Alternatively, the algebraic solution can be written as:

Let the first digit be ‘a’ and the second digit be ‘b’.

10a + b  =  2(a + b)

10a + b  =  2a + 2b

  8a        =   b

Since  a and b are digits, they can only take whole number values between 0 and 9 – this makes the only valid solution to be a = 1 and b = 8.

The algebraic solution has the advantage that it is easily applied to the extension of this problem, i.e. three times, four times, five times or n times the sum of its digits. Can you find the only value of n (between 2 and 9) that has a multiple solution?


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