S={1,2,3,4,5,6,7} f:S→S f(m⋅n)=f(m)f(n) m,n∈S⇒m,n∈S If mn∈S⇒mn≤7 So, (1⋅1,1⋅2,...,1⋅7)≤7 (2⋅2,2⋅3)≤7 When m=1,f(n)=f(1)⋅f(n)⇒f(1)=1 When m=n=2 , f(4)=f(2)f(2)={
f(2)=1⇒f(4)=1 or
f(2)=2⇒f(4)=4.
When, m=2,n=3 f(6)=f(2)f(3){
When, f(2)=1
f(3)=1 to 7
When, f(2)=2
f(3)=1 or 2 or 3.
And f(5),f(7) can take any value (1-7) [∵f(5)=f(1)⋅f(5)≤7 and f(7)=f(1)⋅f(7)≤7} The possible combination is llf(1)=1f(1)=1 f(2)=1f(2)=2 f(3)=(1−7)f(3)=(1−3) f(4)=1f(4)=4 f(5)=(1−7)f(5)=(1−7) f(6)=f(3)f(6)=f(3) f(7)=(1−7)f(7)=(1−7) So, total =(1×1×7×1×7×1×7) +(1×1×3×1×7×1×7) =490