Let $S=\{1,2,3,4,5,6,7\}$. Then the number of possible function $f: S \rightarrow S$ 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 ____.

JEE Main · 2021 · 27 Jul 2021 · Shift 1 · Sets Relations and Functions

Let

S = {1, 2, 3, 4, 5, 6, 7}

. Then the number of possible function

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 ____.

Answer: 490

Taking

m = n = 1

gives

f (1)^{2} = f (1)

, and

f (1) \in S

is nonzero, so

f (1) = 1

; this makes

f (1 \cdot n) = f (1) f (n)

automatic.
The only other products that stay inside

S

are

2 \cdot 2 = 4

and

2 \cdot 3 = 6

, which force

f (4) = f (2)^{2}, f (6) = f (2) f (3)

f (4)

and

f (6)

are determined once

f (2)

and

f (3)

are chosen, subject to

f (2)^{2} \leq 7

and

f (2) f (3) \leq 7

, while

f (5)

and

f (7)

are free.
From

f (2)^{2} \leq 7

f (2) \in {1, 2}

. For

f (2) = 1

every

f (3) \in S

works, giving

7

choices; for

f (2) = 2

the condition

2 f (3) \leq 7

leaves

f (3) \in {1, 2, 3}

, giving

3

choices. Each of

f (5)

and

f (7)

has

7

choices.
The number of such functions is

(7 + 3) \times 7 \times 7 = 490

JEE Main · 2021 · 27 Jul 2021 · Shift 1 · Integer answer