for the solution,
we have been given 7 variables.. {A,B,C,D,E,F,G}
And given domain ={1,2,3,4,5,6,7,8,9}
now we cannot choose values ={5 and 7} as the condition A*B*C=B*G*F=D*E*F cannot be satisfied... because atleast one row or column might not fulfill the condition... = 5answer
.
now for values of given variables...
so required integer values = {1,2,3,4,6,8,9}
now we have = A*B*C*D*E*F*G = 1*2*3*4*6*8*9 = 2^7*3^4
= (A*B*C)^2 * G = 2^7 * 3^4 (as ABC=DEF)
NOW FROM THIS FOUR VARIABLES ONE MUST BE 2.
And if we choose any one from A,B,C to be 2 than G will be a multiple of 2^5 which will not be in the given domain.
therefore, G=2.
now (A*B*C)^2 = (2^6 * 3^4)
A*B*C = 2^3 * 3^2 = 72
= 9*8*1 or 3*4*6
ALSO, B*G*F = 72 (AS G=2)
B*F = 36 (9*4)
SO , the given solution would look like
or , you can inter-change the values between 1,8 and 3,6 or 9,8(such that ABC=BGF=DEF)=72
and this can be done in 8ways...