Try this beautiful problem based on Pairs of Positive Integer, useful for ISI B.Stat Entrance.
How many pairs of positive integers (m, n) are there satisfying \(m^3 – n^3 = 21\)?
Integer
Factorization
Odd number
Answer: (c) is an integral multiple of 6
TOMATO, Problem 156
Challenges and Thrills in Pre College Mathematics
Given that \(m^3– n^3 = 21\)
\(\Rightarrow (m – n)(m^2 + mn + n^2) = 3*7\)
Possible cases are , \(m – n = 3\), \(m^2 + mn + n^2 = 7\) and \(m – n = 1\), \(m^2 + mn + n^2 = 21\)
Can you now finish the problem ..........
Now according to the questions we are trying to find out the positive integers,so we will neglect the negetive cases..........
First case, \((3 + n)^2 + (3 + n)n + n^2 = 7\)
\(3 n^2 + 9n + 2 = 0\)
\(n = \frac {-9 \pm \sqrt {9^2 – 432}}{6}\)= not integer solution.
So this case is not possible.
Second case, \((n + 1)^2 + (n + 1)n + n^2 = 21\)
\(\Rightarrow 3n^2 + 3n – 20 = 0\)
\(n = \frac {-3 \pm \sqrt {9 + 240}}{6}\) = not integer solution.
Therefore option (b) is the correct
AMC - AIME Boot Camp... for brilliant students.
Use our exclusive one-on-one plus group class system to prepare for Math Olympiad
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies. The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are […]
Cheenta Academy proudly celebrates the success of 27 current and former students who qualified for the Indian Olympiad Qualifier in Mathematics (IOQM) 2025, advancing to the next stage — RMO. This accomplishment highlights their perseverance and Cheenta’s ongoing mission to nurture mathematical excellence and research-oriented learning.