@ ALL :

A LITTLE CORRECTION :

In the posted solution a little correction regarding the notation is needed.

Please refer to the 4th line of the solution ,i.e , the following statement :

" where 2 ,3,p1,p2,p3,p4,p5,....,pn are the PRIME factors of N and (a1)+1 ,(a2)+2 ,a3 , a4 ,a5 ,....,an are the powers of them respectively and so each of (a1),(a2),a3 , a4 ,a5 ,....,an is ofcourse non-negative +ve integer."

----Here in the 1st line p1 and p2 have been typed by mistake though these two

notations are absent in the whole solution and therefore p1 and p2 should be

deleted from the above statement,consequently the statement should be read as :

" where 2,3,p3,p4,p5,....,pn are the PRIME factors of N and (a1)+1 ,(a2)+2 ,a3 , a4 ,a5 ,....,an are the powers of them respectively and so each of (a1),(a2),a3 , a4 ,a5 ,....,an is ofcourse non-negative +ve integer."

If u have any confusion or complaint regarding the posted solution along with

the correction, and also any difficulty regarding the understanding of the

solution then u are earnestly requested to communicate it with me..

Your feedback is highly solicited.

Thanks to all:-) :-)

According to the general prime factorization of a natural number We can factor the given integer N as follows.

Let, N = 18*{2^(a1)}*{3^(a2)}*{(p3)^(a3)}*{(p4)^(a4)}*{(p5)^(a5)}....{(pn)^(an)}

=>N={2^[(a1)+1]}*{3^[(a2)+2]}*{(p3)^(a3)}*{(p4)^(a4)}*{(p5)^(a5)}...{(pn)^(an)}

where 2 ,3,p1,p2,p3,p4,p5,....,pn are the PRIME factors of N and (a1)+1 ,(a2)+2 ,a3 , a4 ,a5 ,....,an are the powers of them respectively and so each of (a1),(a2),a3 , a4 ,a5 ,....,an is ofcourse non-negative +ve integer.

So, from above we can write that

(the number of divisors of N) = {(a1)+2}*{(a2)+3}*(a3 +1)*(a4 +1)*....(an +1)

==> 10 = {(a1)+2}*{(a2)+3}*(a3 +1)*(a4 +1)*....(an +1)

==> 2 *5 = {(a1)+2}*{(a2)+3}*(a3 +1)*(a4 +1)*....(an +1)......[(eqn1)]

Now, since each of (a1),(a2),a3 , a4 ,a5 ,....,an is ofcourse non-negative +ve integer , so , clearly

{(a1)+2} > or, = 2;
and
{(a2)+3} > or, = 3;

Now, at this situation it clearly implies that
a3=0; a4=0; a5=0; .... ; an=0.
and
{(a1)+2} = 2 ; ==> a1 = 0 ;

{(a2)+3} = 5 ; ==> a2 = 2 ;

So, the integer
N = {2^[(a1)+1]}*{3^[(a2)+2]}*{(p3)^(a3)}*{(p4)^(a4)}*{(p5)^(a5)}...{(pn)^(an)}

=> N = (2^1)*(3^4);
=2*81
=162.

So, N=162..