The numbers with HCF 36 will be

2^3 x 3^2 x 7^2; number of factors (3+1)(2+1)(2+1)=36
2^2 x 3^3 x 5^2; number of factors (2+1)(3+1)(2+1)=36

Least LCM will be 2^3 x 3^3 x 5^2 x 7^2

Two numbers have 36 factors each and HCF of these two numbers is 1036. What is the minimum possible LCM of these two numbers if the power of any prime factor in these two numbers is not more than 3?

had this question been like:
factors of 1036 = 2^2 x 7 x 37
let the two numbers be 1036 a and 1036 b
these two numbers cannot have any other new common prime factor otherwise HCF will change

Now we want each number to have 36 factors (3x3x2x2) or (4x3x3) other combinations will not be allowed as it will make the power of any prime number to go over 3.

(4x3x3) means these 2^3 x 7^2 x 37^2 ; 2^2 x 7^3 x 37^2 ; 2^2 x 7^2 x 37^3
(3x3x2x2) means one of the number can be 2^2 x 3^2 x 7 x 37 ; 2^2 x 5^2 x 7 x 37 [others will have larger values]

from the above numbers the lower two will give least LCM

So for minimum possible LCM a can be 3^2 and b can be 5^2

LCM will be 2^2 x 3^2 x 5^2 x 7 x 37

I don't know how to explain it in easier way, if anyone else can help...........

these two numbers cannot have any other new common prime factor otherwise HCF will change

36 factors (3x3x2x2) or (4x3x3) other combinations will not be allowed as it will make the power of any prime number to go over 3.

from the above numbers the lower two will give least LCM

Good that you understood because I am no expert at explaining or finding a shorter approach. I just got started with my preparation. Actually I have few Career Launcher booklets and also Math book by HC Verma (not that good a book but some fundas are good), while coming to US what ever I could get hold of I packed. Sorry no good link that I know of.

Two numbers have 36 factors each and HCF of these two numbers is 36. What is the minimum possible LCM of these two numbers if the power of any prime factor in these two
numbers is not more than 3?

cube(2) X cube(3) X sqr(5) X 7
sqr(2) Xsqr( 3) X cube(5) X cube(7)
cube(2) Xcube( 3) X sqr(5) Xsqr( 7)
cube(2)X 32 X cube(5) X sqr(7)
cube(2) X sqr(3) X sqr(5) X sqr(7)