Let S= 1,2,3,4,5,6,7 . Then the number of possible functions f: S arrow S such that f(m ⋅ n)=f(m) ⋅ f(n) for every m, n ∈ S and m ⋅ n ∈ S is equal to.

Question

Let S= 1,2,3,4,5,6,7 . Then the number of possible functions f: S arrow S such that f(m ⋅ n)=f(m) ⋅ f(n) for every m, n ∈ S and m ⋅ n ∈ S is equal to. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

F(m n)=f(m) ⋅ f(n) Put m=1 f(n)=f(1) ⋅ f(n) ⇒ f(1)=1 Put m=n=2 f(4)=f(2) ⋅ f(2) begincasesf(2)=1 ⇒ f(4)=1 text or f(2)=2 ⇒ f(4)=4 endcases Put m=2, n=3 f(6)=f(2) ⋅ f(3) begincases text when f(2)=1 f(3)=1 text to 7 f(2)=2 f(3)=1 text or 2 text or 3 endcases f(5), f(7) can take any value Total =(1 × 1 × 7 × 1 × 7 × 1 × 7)+(1 × 1 × 3 × 1 × 7 × 1 × 7) =490

Q. Let S={1,2,3,4,5,6,7}. Then the number of possible functions f:S→S such that f(m⋅n)=f(m)⋅f(n) for every m,n∈S and m⋅n∈S is equal to____.

Q. Let $S = {1, 2, 3, 4, 5, 6, 7}$ . Then the number of possible functions $f : S \to S$ such that $f (m \cdot n) = f (m) \cdot f (n)$ for every $m, n \in S$ and $m \cdot n \in S$ is equal to____.