Smallest value | PRMO 2018 | Question 15

Content

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?

Algebra

Numbers

Multiples

But try the problem first…

Answer: is 21.

Source

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

2a-b=\(k_2^2+k_3^2\)

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.