Let $AP(a,d)$ denote the set of all the term of an infinite arithmetic progression with first term a and common difference d>0. If $AP(1,3)\ \cap AP(2,5)\ \cap AP(3,7)=AP(a,d)$ then find the value of a+d.
My approach
$1 + \left( {{n_1} - 1} \right)3 = X$
$2 + \left( {{n_2} - 1} \right)5 = X$
$3 + \left( {{n_3} - 1} \right)7 = X$
$1 + \left( {{n_1} - 1} \right)3 = 2 + \left( {{n_2} - 1} \right)5 = 3 + \left( {{n_3} - 1} \right)7 \Rightarrow 3{n_1} - 2 = 5{n_2} - 3 = 7{n_3} - 4 = X$
${n_1} = \frac{{X + 2}}{3};{n_2} = \frac{{X + 3}}{5};{n_3} = \frac{{X + 4}}{7}$
We need to find the value where $n_1, n_2,n_3$ are integer and through excel I found the first term as 52.
How do we calculate the first term a=52, d=LCM(3,5,7)=105