<s>[INST] How many factors does 6N have? Given a number $2N$ having 28 factors another number $3N$ having 30 factors, then find out the number of factors of $6N$.  
 [/INST] If $2n$ has 28 divisors, then $2n$ is of the form $a^6.b$, where b has 4 divisors.  Note that if chose $2^13$ or $2^26$ here, then the third multiple would have more than $30$ divisors, since it would be $3*13* or $2*27$ divisors required.  
Since $3n$ does not have a divisor of the form $a^6$, we see that  $a=2$, and that n is at least $2^5$.  The reminder takes $b$ has $4$ divisors, and 3b has $5$, so $ b=3^3$. So the number is $864$, which has $6*4=24$ divisors, its second and third have $28$ and $30$ divisors.
Likewise $6n$ has 7*5 or $35$ divisors.  It is the $5184$.
One can also see that if $6n=2^a 3^b c$, then for $2n$ to have 28 divisors, and $3n$ to have 30 divisors, then $7|a-1$ and $5|b-1$.  This, and that $ab+b=28$ and $ab+a=30$, means that $a-b=2$, then $a=6, b=4$ is the only solution here.  
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