# In two digit number,…

In two digit number, the product og digits is $20$ and if $9$ is added to the number, the digits will be reversed. What is the number?

Can anyone help me with this? Thanks.

• How many different ways can $20$ be written as the product of two single-digit numbers? – John Wayland Bales Jan 8 '17 at 2:11
• @John Wayland Bales, $4*5$. – user292114 Jan 8 '17 at 2:12
• So do you see what the answer must be now? – John Wayland Bales Jan 8 '17 at 2:14
• @John Wayland Bales, yea. – user292114 Jan 8 '17 at 2:15

There is only one possible combination of two single digits that give $20$ when multiplied together. They can be put after one another in two different ways. One of those ways makes the resulting number fulfill the second criterion, the other does not.

Let the two digit number be $xy=10x+y$. Then your conditions mean that:

• $xy=20$ and
• $10x+y+9=10y+x \Leftrightarrow x+1=y$

Can you solve the above for $x,y$ non-zero digits?

$ab=20$.

(Product of digits)

$10a+b+9=10b+a$.

(First digit * 10 (place value) + Second digit = Number)

$9a+9= 9b$

$a+1=b$.

(Simplification)

We can see that |ab| is 45.

* I neglected the case where b = 0. That would mean adding 9 makes b=9. However, it is negligible because if b were 0, that would mean the product is 0, which goes against the product being 20 *