JEE Advanced 2019 — Paper 1 — Question 45

$\operatorname{AP}(1,3) \equiv\{1,4,7,10, \ldots\}=.\{n / n=3 k+1, k \in W\}$

$\operatorname{AP}(2,5) \equiv\{2,7,12, \ldots\}=.\{n / n=5 k+2, k \in W\}$

$\operatorname{AP}(3,7) \equiv\{3,10,17, \ldots\}=\{n / n=7 k+3, k \in W\}$

Let common term is M

$$M \equiv 1(\bmod 3), M \equiv 2(\bmod 5), M \equiv 3(\bmod 7)$$

$\Rightarrow \quad \mathrm{M} \equiv 52(\bmod 105)$

so $\quad a=52, d=105$ and $a+d=157$