Let S = \ 1, 2, 3, 4, 5, 6, 7\ . Then the number of possible...

Given,

$f\left(mn\right) = f\left(m\right) \cdot f\left(n\right)$

Put $m = 1, then$f\left(1 \cdot n\right) = f\left(1\right) \cdot f\left(n\right) ⇒ f\left(1\right) = 1\left\{f\left(n\right) \ne 0\right\}$</p><p>Put$m = n = 2$</p><p>$f\left(4\right) = f\left(2\right) \cdot f\left(2\right)$</p><p>If$f\left(2\right) = 1,$then$f\left(4\right) = 1$and if$f\left(2\right) = 2,$then$f\left(4\right) = 4$</p><p>Put$m = 2, n = 3$</p><p>$f\left(6\right) = f\left(2\right) \cdot f\left(3\right)$</p><p>If$f\left(2\right) = 1,$then$f\left(3\right)$can be any value between$1$to$7$</p><p>If$f\left(2\right) = 2,$then$f\left(3\right)$can be$1$or$2$or$3$</p><p>Similarly$f\left(5\right), f\left(7\right)$can take any value from \left\{1, 2, . . . , 7\right\}$

So, total number of function $= \left(1 \times 1 \times 7 \times 1 \times 7 \times 1 \times 7\right) + \left(1 \times 1 \times 3 \times 1 \times 7 \times 1 \times 7\right)$

$= 490$