# A number of 2 digits is 6 times the sum of its digits. If 9 is substracted from the number, the numbers reverse. Find the numbers.

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### 1 Answer

Let `n=10a+b` be your number, where `a` is the tens digit and `b` is the ones digit.

Then since the number is 6 times the sum of its digits, we get the equation

`10a+b=6(a+b)`

this can be expanded on the RHS, then simplified to get

`10a+b=6a+6b`

`4a=5b`

Let's call this equation (1).

Now if we subtract 9 from the number, we get the digits reversed.

This is the same as saying that:

`10a+b-9=10b+a`

Which can simplify to

`9a-9b-9=0`

Upon dividing by 9, and multiplying by 4 we get

`4a-4b-4=0`

We call this equation (2).

To find the number N, we now need to solve the linear system of equations (1), (2).

Substitute (1) directly into (2) to get

`5b-4b-4=0`

After collecting like terms, this means that `b=4` .

Substitute into (1) to get `a=5`.

As a check that the number is N=54, we see that the sum of the digits is 9, and `6\times 9=54`. Also, `54-9=45`, which is the original number with digits reversed.` `

**The number you are looking for is 54.**