Question 565018
If two of the positive factors of the number are
{{{6=2*3}}} and {{{25=5^2}}} ,
the number must be a multiple of {{{2*3*5^2=150}}}
The number 150 has {{{2*2*3=12}}} positive factors.
Any other number who has 6 and 25 as factors will be a multiple of 150, and will have more than 12 factors.
So your locker is number 150.
How do I know {{{2*3*5^2=150}}} has {{{2*2*3=12}}} positive factors?
Because all the factors will be of the form
{{{2^a*3^b*5^c}}} where a and b could be 0 or 1, and c could be 0, 1, or 2.
That gives you 2 choices for a, 2 choices for b,and 3 choices for c. That makes  {{{2*2*3=12}}} combinations.
In case you don't believe, I'll list the factors of 150
{{{1=1*1*1=2^0*3^0*5^0}}} {{{2=2*1*1=2^1*3^0*5^0}}}
{{{3=1*3*1=2^0*3^1*5^0}}} {{{6=2*3*1=2^1*3^1*5^0}}}
{{{5=1*1*5=2^0*3^0*5^1}}} {{{10=2*1*5=2^1*3^0*5^1}}}
{{{15=1*3*5=2^0*3^1*5^1}}} {{{30=2*3*5=2^1*3^1*5^1}}}
{{{25=1*1*25=2^0*3^0*5^2}}} {{{50=2*1*25=2^1*3^0*5^2}}}
{{{75=1*3*25=2^0*3^1*5^2}}} {{{150=2*3*25=2^1*3^1*5^2}}}