# 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.

Asked on by spsaroj

lfryerda | High School Teacher | (Level 2) Educator

Posted on

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.

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