If four digit numbers are formed by using the digits $1,2,3,4,5,6,7$ without repetition, then out of these numbers, the numbers exactly divisible by 25 are

For a four-digit number formed from the digits $\{1,2,3,4,5,6,7\}$ to be divisible by 25, its last two digits must form a number divisible by 25.

The possible two-digit endings are $25$ and $75$, as $00$ and $50$ are not feasible without the digit $0$.

Case ending in 25: The digits 2 and 5 are used, leaving $\{1,3,4,6,7\}$ for the first two positions. There are $5 \times 4 = 20$ arrangements.

Case ending in 75: The digits 7 and 5 are used, leaving $\{1,2,3,4,6\}$ for the first two positions. There are $5 \times 4 = 20$ arrangements.

Total numbers: $20 + 20 = 40$.

The number of valid four-digit numbers divisible by 25 is $\boxed{40}$.