<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] Let $p_i$, be primes in increasing order. Consider the prime factorization of $n$. 
$$ n = 2^{a_1} \times 3^{a_2} \times \prod p_i^{a_i}, $$
where $a_i$ are non-negative integers. For this question, let the indexing run over $i\geq 3$.
You are given that 
$$ (a_1 + 2) \times (a_2 +1 ) \times \prod (a_i + 1) = 28 $$
and
$$( a_1  + 1) \times (a_2 + 2) \times \prod(a_i + 1) = 30 $$
The factors of 28 are $1, 2, 4, 7, 14, 28$. The factors of 30 are $1, 2, 3, 5, 6, 10, 15, 30$.
A quick check shows that the possible values of $a_1$ are 2 or 5.    
If $a_1 = 2$, then we must have 
$$ \frac{ a_2 +1 } { a_2 + 2} = \frac{7}{10},$$
which has no positive integer solution.
If $a_1 = 5$, then we must have 
$$\frac{a_2 + 1} { a_2 + 2} = \frac{4}{5}, $$
which has a positive integer solution $a_2 = 3$. This implies that $\prod (a_i+1) = 1$. Hence, the number of factors of $6n$ is 
$$( a_1  + 2) \times (a_2 + 2) \times \prod(a_i + 1) = 7 \times 5 \times 1 = 35 $$
In this case, $n = 2^5 \times 3^3$.
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