Find the number of pairs (a, b) of natural numbers such that b is a 3 -digit number, a+1 divides b-1 and b divides a^2+a+2 .

Step-by-step solution

aligned & 100 b 999 & (b-1)= (a+1)....(i) aligned and a^2+a+2= b ...(ii) a^2+a+2=b, 2 b, 3 b, _ _ _ _ i.e., a 14 hence for a=14 gives a^2+a+2=212 i.e., b=106,212 but (a, b)=(14,106) satisfy (i) Similarly (16,137),(18,171) _ _ _ _(44,991) as b 999 . Hence, 16 orderd pairs.