Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Basics of Mathematics

Olympiad workbook · IOQM · Basics of Mathematics

Find the number of ordered pairs (a, b) such that a, b ∈\ 10,11, ,...

Find the number of ordered pairs (a,b)(a, b) such that a,b∈{10,11,…,29,30}a, b \in\{10,11, \ldots, 29,30\} and GCD⁡(a,b)+LCM⁡(a,b)=a+b\operatorname{GCD}(a, b)+\operatorname{LCM}(a, b)=a+b.

Correct answer

35

Step-by-step solution

g+l=a+bg+abg=a+bg2+(a+b)g+ab=0(g−a)(g−b)=0g=a,b\begin{aligned} & g+l=a+b \\ & g+\frac{a b}{g}=a+b \\ & g^2+(a+b) g+a b=0 \\ & (g-a)(g-b)=0 \\ & g=a, b \end{aligned} For a=ba=b, there will be 21 cases. If a=10,ba=10, b may be 20 or 30 as well and vice-versa. \left.a=11,b=22a=12,b=24a=13,b=26a=14,b=28a=15,b=30\begin{array}{ll}a=11, & b=22 \\ a=12, & b=24 \\ a=13, & b=26 \\ a=14, & b=28 \\ a=15, & b=30\end{array}\right] \times 2=10$ ways
Practice Basics of Mathematics on Quantrex Academy →

More from Basics of Mathematics

All Basics of Mathematics questions Full Basics of Mathematics list All Olympiad workbook PYQs