Let a ( 120 ) = f1 * f2 * f3 * f4 * f5 .... ( 2 * 2 * 2 * 3 * 5 ).
b ( 18 ) = F1 * F2 * F3........ ( 2 * 3 * 3 )
LCM of a( 120 ) and b( 18 ) = product of elements of a union b.
= M( {f1, f2, f3, f4, f5 } U { F1, F2, F3 } ) //M- > multiplication.
= M( (f1/F1), f2, f3, (f4/F2), F3, f5 ) ----- (A) // f1/ F1- > f1 or F1
HCF of a( 120 ) and b( 18 ) = product of elements of a intersection b.
= M( {f1, f2, f3, f4, f5 } $\cap$ { F1, F2, F3 } )
= M( (f1/F1), (f4/F2) ) -----------------------(B)
// Remember if a $\cap$ b= ∅, then HCF = 1.
From equation A and B,
Product of LCM and HCF = M( (f1/F1), f2, f3, (f4/F2), F3, f5 ) * M( (f1/F1), (f4/F2) )
= M( f1, f2, f3, f4, F3, f5, F1, F2 )
= Product of a and b.