What is the smallest two-digit integer $n$ such that switching its digits and then adding 3 results in $2n$

We can write

n = 10a + b

where a and b belong to {1, 2, 3, …, 9}.

Swapping the digits produces a new number

n' = 10b + a

and adding 3 to this number makes up 2n,

n' + 3 = 2n

or

(10b + a) + 3 = 2 (10a + b)

so that

19a - 8b = 3

Since we want n to be the smallest possible 2-digit number that meets the criteria, let's start with a = 1. Then solving for b, we find

19 - 8b = 3

16 = 8b

b = 2

So, the number we're looking for is n = 12.

Just to confirm, swapping the digits gives 21, and 21 + 3 = 24 = 2•12.

step-by-step explanation:

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