The average of seven consecutive positive integers is $126$. The smallest of these integers is :

For an odd number of consecutive integers, the average is always equal to the exact middle term.
Since there are $7$ terms, the middle term is the $4^{\text{th}}$ term.

Therefore:

$$\text{Middle term }(4^{\text{th}}\text{ term})=126$$

Since the numbers are consecutive integers, we count backward to find the smallest ($1^{\text{st}}$) integer:

* $3^{\text{rd}}$ term = $126-1=125$
* $2^{\text{nd}}$ term = $125-1=124$
* $1^{\text{st}}$ term = $124-1=123$

Alternatively, let the smallest integer be $x$. The $7$ consecutive integers are $x,x+1,x+2,x+3,x+4,x+5,x+6$.

$$\text{Average}=\frac{7x+(1+2+3+4+5+6)}{7}=\frac{7x+21}{7}=x+3$$

Given that the average is $126$:

$$x+3=126⟹x=123$$

Thus, the smallest integer is $123$.