@Konda and Monil... for the problems where we are needed to find all sets of numbers which give a lcm of a certain number, i tried some approach and looks like this can be generalised and used until there is some better approach posted..
now let N = 12 = 2^2*3
now the no of factors of 12 = 3*2 = 6
all these 6 factors will give the LCM 12 with the number 12,
apart from these we have the co-primes, we will take one at a time, also to get the LCM, we need the highest power of the factor in the number,
so (2^2,3) also (2^2,6)
so the total un-ordered pairs of nos = 6+2 = 8
If the ordered pairs are asked then = 8*2-1 =15 {(12,12) counted twice)
let us consider for some bigger no, n =60 = 2^2*3*5
no that have lcm 60 wit 60 = 3*2*2 = 12
co-primes: (2^2*15), -- 2 terms
(12,5) -- 1 term
(20,3) -- 1 term
no of un-ordered terms --- 16
no of ordered ----- 31 (excluding (60,60) which is counted twice)
for the given problem,
2^3.5^7.11^13no of factors that have lcm as 2^3.5^7.11^13 with 2^3.5^7.11^13 = 4*8*14 = 448
co-primes: (2^3, 5^7.11^13) -- 3 terms
(5^7, 2^3*11^13) ---- 7 terms
(11^13, 2^3*5^7) ---- 11 terms
so un-ordered terms = 448+21 = 469 terms...
ordered pairs = 937 pairs
Is this answer correct?? I think it is right..

I think u can identify a pattern in all these questions, and if u need to generalise using a formula, then it is, (for n=a^p.b^q.c^r)
(p+1)(q+1)(r+1) +(p+q+r)...for un-ordered pairs
I hate formulas...The more I know the less confident I am... (may be my next signature..

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