Let $\mathrm{A}\mathrm{P}\mathrm{ }\left(\mathrm{a};\mathrm{d}\right)$ denote the set of all the terms of an infinite arithmetic progression with first term a and common difference $\mathrm{d}>0.$ If $\mathrm{A}\mathrm{P}\left(1;3\right)\cap \mathrm{A}\mathrm{P}\left(2;5\right)\cap \mathrm{A}\mathrm{P}\left(3;7\right)=\mathrm{A}\mathrm{P}\left(\mathrm{a};\mathrm{d}\right)$ then $\mathrm{a}+\mathrm{d}$ equals ____

Let $T_{\left(1, m\right)}:$ $m^{\mathrm{t}\mathrm{h}}$ term of $1^{\mathrm{s}\mathrm{t}}$ series $A\left(1;3\right)$
$T_{\left(2, n\right)}:$ ${\mathrm{n}}^{\mathrm{t}\mathrm{h}}$ term of $2^{\mathrm{n}\mathrm{d}}$ series $A\left(2;5\right)$
$T_{\left(3, r\right)}:$ $r^{\mathrm{t}\mathrm{h}}$ term of $3^{\mathrm{r}\mathrm{d}}$ series $A(3;7)$
For common terms
$T_{(1, m)}=T_{(2, n)}=T_{(3, r)}$
$1+3\left(m-1\right)=2+5\left(n-1\right)=3+7\left(r-1\right)$
$3m-2=5n-3=7r-4$
$\Rightarrow m=\frac{5n-1}{3}\mathrm{a}\mathrm{n}\mathrm{d}r=\frac{5n+1}{7}$
For $‘m’$ to be a natural number $n=\mathrm{2,5},11,\dots$
And for $‘r’$ to be a natural number $n=\mathrm{4,11},\dots$
When $n=11,m=18$ and $r=8$
$\Rightarrow$ For $1^{\mathrm{s}\mathrm{t}}$ common term of three series $A(a,d)$
$a=1^{\mathrm{s}\mathrm{t}}$ term of $A\left(\mathrm{1,3}\right)$
$a=1+\left(181\right)3=52$
$\Rightarrow$ Common difference of $A\left(a,d\right)=\mathrm{L}\mathrm{C}\mathrm{M}$ of common difference of $\left(A\left(1;3\right),A\left(2;5\right),A\left(3;7\right)\right)$
$d=\mathrm{L}\mathrm{C}\mathrm{M} (3, 5, 7)$
$d=105$
Hence, $a+d=52+105=157$