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.
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
this can be expanded on the RHS, then simplified to get
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:
Which can simplify to
Upon dividing by 9, and multiplying by 4 we get
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
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.