Let say $3x,4x,5x,7x$ are the four parts of the $152$
we can write like this $3x+4x+5x+7x = 152$
$\implies 19x=152$
$\implies x = 8$
So, four parts is $: 3\times 8,4\times 8,5\times 8,7\times 8 = 24,32,40,56$
So, the smallest part is $24.$
Option $(D)$ is the correct answer.