jmc

The desired integer has at least two digits. Let $d$ be its leftmost digit, and let $n$ be the integer that results when $d$ is deleted. Then for some positive integer $p$, $10^p\cdot d+n=29n$, and so $10^p\cdot d=28n$. Therefore 7 is a divisor of $d$, and because $1\le d\le 9$, it follows that $d=7$. Hence $10^p=4n$, so $n=\frac{10^p}{4}=\frac{100\cdot 10^{p-2}}{4}=25\cdot 10^{p-2}$. Thus every positive integer with the desired property must be of the form $7\cdot 10^p+25\cdot 10^{p-2}=10^{p-2}(7\cdot 10^2+25)=725\cdot 10^{p-2}$ for some $p\ge 2$. The smallest such integer is $\boxed{725}$.