Question 990263
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<B>Answer</B>. 36.


<B>Solution</B>


Let &nbsp;<B>a</B>&nbsp; and &nbsp;<B>b</B>&nbsp; be the first &nbsp;(the left)&nbsp; and the second &nbsp;(the right)&nbsp; digit of the number respectively, &nbsp;so that the number is &nbsp;10a + b. 


The first condition says that &nbsp;{{{(10a + b)/(ab)}}} = {{{2}}}, &nbsp;&nbsp;or 


10a + b = 2ab. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1)


The number with swapped digits is &nbsp;10b + a, so the second condition says that


10a + b + 27 = 10b + a, &nbsp;&nbsp;or &nbsp;&nbsp;9a - 9b = -27, &nbsp;&nbsp;or &nbsp;&nbsp;a-b = -3.


So, &nbsp;we have a system of two equations 


{{{system(10a + b = 2ab,
a-b = -3)}}}.


Express &nbsp;<B>b</B>&nbsp; from the last equation, &nbsp;b = a + 3, &nbsp;and substitute it into the previous equation. &nbsp;You will get


10a + a + 3 = 2a*(a+3), &nbsp;&nbsp;or


{{{2a^2}}} - {{{5a}}} - {{{3}}} = {{{0}}}.


Solve this quadratic equation using the quadratic formula. &nbsp;You will get the roots &nbsp;{{{a[1]}}} = {{{3}}} &nbsp;and&nbsp; {{{a[2]}}} = {{{-1/2}}}. 

The negative root doesn't suit, &nbsp;because we are looking for the digit, &nbsp;which should be non-negative integer less than 10. &nbsp;The root &nbsp;{{{a[1]}}} = 3 &nbsp;is good. 


So, &nbsp;the number is &nbsp;36.


Please check yourself that this number satisfies all conditions of the problem.