Question 1151077
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The basic problem here is a common one.  With the two poles of heights a and b, the height above the ground at which the two wires cross is {{{ab/(a+b)}}}<br>
So in this problem we have<br>
{{{ab/(a+b)=10}}}<br>
and we need to determine the number of possible values for a if a and b are both integers.<br>
This is a Diophantine equation -- we have two variables but only one equation; but the number of solutions is limited by the requirement that both variables be integers.<br>
The following is one standard way for solving such equations.<br>
(1) Solve the equation for one variable in terms of the other.<br>
{{{ab/(a+b) = 10}}}
{{{ab = 10a+10b}}}
{{{ab-10b = 10a}}}
{{{b(a-10) = 10a}}}
{{{b = 10a/(a-10)}}}
{{{b = (10a-100+100)/(a-10)}}}
{{{b = (10a-100)/(a-10)+100/(a-10)}}}
{{{b = 10+100/a-10}}}<br>
(2) b and 10 are integers; that means {{{100/(a-10)}}} must be an integer.  And that means (a-10) must be a factor of 100.<br>
We could identify all the possible values of a; but the problem only asks for the number of possible values for a.  100 has 9 positive factors, so there are 9 possible values for a.<br>
ANSWER:  There are 9 possible heights for pole A.<br>