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Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Basics of Mathematics

Olympiad workbook · IOQM · Basics of Mathematics

Find the number of pairs (a, b) of natural numbers such that b is a...

Find the number of pairs (a,b)(a, b) of natural numbers such that bb is a 3 -digit number, a+1a+1 divides b−1b-1 and bb divides a2+a+2a^2+a+2.

Correct answer

16

Step-by-step solution

100≤b≤999(b−1)=λ(a+1)....(i)\begin{aligned} & 100 \leq b \leq 999 \\ & \quad(b-1)=\lambda(a+1)....(i) \end{aligned} and a2+a+2=μba^2+a+2=\mu b...(ii) ∴a2+a+2=b,2b,3b,\therefore a^2+a+2=b, 2 b, 3 b, ____\_\_\_\_ i.e., a≥14a \geq 14 hence for a=14a=14 \quad gives a2+a+2=212a^2+a+2=212 i.e., b=106,212b=106,212 but (a,b)=(14,106)(a, b)=(14,106) satisfy (i) Similarly (16,137),(18,171)(16,137),(18,171) ____\_\_\_\_ (44,991)(44,991) as b≤999b \leq 999. Hence, 16 orderd pairs.
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