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# 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

Algebra

Numbers

Multiples

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

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.