Question 210031
what is the solution of,  a two digit number is four times the sum of its digits and thrice the product of its digits.  find the number.
<pre><font size = 4 color = "indigo"><b>
t = tens digit
u = ones digit
the two digit number = 10t+u
t+u = sum of the digits
tu = product of the digits
</b></pre></font>
>>...A two digit number is four times the sum of its digits...<<
<pre><font size = 4 color = "indigo"><b>
So 

{{{10t+u=4(t+u)}}}

Distribute on the right

{{{10t+u=4t+4u}}}

Simplify:

{{{6t=3u}}}

Divide both sides by 3

{{{2t=u}}}
</b></pre></font>
>>...a two digit number is...thrice the product of its digits...<<
<pre><font size = 4 color = "indigo"><b>
{{{10t+u=3(tu)}}}

{{{10t+u=3tu}}}

So we have this system of equations:

{{{system(2t=u,10t+u=3tu)}}}

Substitute {{{2t}}} for {{{u}}} in the 2nd equation

{{{10t+(2t)=3t(2t)}}}

{{{10t+2t=6t^2}}}

Combine terms on the left

{{{12t=6t^2}}}

Divide through by 6

{{{2t=t^2}}}

{{{0=t^2-2t}}}

{{{t^2-2t=0}}}

Factor out {{{2}}}

{{{t(t-2)=0}}}

Set each factor = 0

{{{t=0}}}   {{{t-2=0}}}
            {{{t=2}}} 

The ten's digit cannot be 0 for a 2-digit number.

So  {{{t=2}}}

Substitute that in {{{2t=u}}}

{{{2t=u}}}

{{{2(2)=u}}}

{{{4=u}}}

So since {{{t=2}}} and {{{u=4}}},

The number is {{{24}}}.

Checking:

The sum of the digits is 2+4 or 6

So 24 is 4 times the sum of its digits, 
because 24 is 4 times 6.

The product of the digits is 2×4 = 8

So 24 is thrice the product of its digits,
because 24 is 3 times 8.

It checks.  

Edwin</pre>