How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
Learn More

Smallest value | PRMO 2018 | Question 15

Try this beautiful problem from the PRMO, 2018 based on Smallest value.

Smallest Value - PRMO 2018

Let a and b natural numbers such that 2a-b, a-2b and a+b are all distinct squares. What is the smallest possible value of b?

  • is 107
  • is 21
  • is 840
  • cannot be determined from the given information

Key Concepts




Check the Answer

Answer: is 21.

PRMO, 2018, Question 15

Higher Algebra by Hall and Knight

Try with Hints

First hint

2a-b=\(k_1^2\) is equation 1

a-2b=\(k_2^2\) is equation 2

a+b=\(k_3^2\) is equation 3

Second Hint

adding 2 and 3 we get


or, \(k_2^2+k_3^2\)=\(k_1^2\) \((k_2<k_3)\)

Final Step

For least 'b' difference of \(k_3^2\) and \(k_2^2\) is also least and must be multiple of 3

or, \(k_2^2\)=a-2b=\(9^2\) and \(k_3^2\)=a+b=\(12^2\)

or, \(k_3^2-k_2^2\)=3b=144-81=63

or, b=21

or, least b is 21.

Subscribe to Cheenta at Youtube

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.