Question

triplets:- (3,4,5), (6,7,8), (10,24,26), (2,3,4)

solve it with this method:-
if 'a','b' and 'c' are three natural numbers with 'a' as the smallest of them, then,

i) if 'a' is odd, sum of other two numbers is a² and their difference is 1.
ii) if 'a' is even, sum of other two numbers is a²/2 and their difference is 2.​

Answer 1

$\large{\text{\underline{Question:}}}$

Find the triplets $(a,b,c)$ such that if $a$ is odd, the sum of the other two numbers is $a^{2}$ and their difference is $1$, and if $a$ is even, the sum of the other two numbers is $\dfrac{a^{2}}{2}$ and their difference is $2$.​ $a$ is the least natural number in the triplet.

$\large{\text{\underline{To find:-}}}$

A triplet $(a,b,c)$ that satisfies the given condition.

$\large{\text{\underline{Solution:-}}}$

$\underline{\text{Step A: Finding the triplet when }a\text{ is odd.}}$

If $a$ is odd, and the sum of the other two numbers is $a^{2}$ and their difference is $1$, it gives the conditions,

$\hookrightarrow b+c=a^{2},\ c-b=1$

By solving the equation,

$\hookrightarrow b=\dfrac{a^{2}-1}{2} ,\ c=\dfrac{a^{2}+1}{2}$

$\underline{\text{Step B: Finding the triplet when }a\text{ is even.}}$

If $a$ is even, and the sum of the other two numbers is $\dfrac{a^{2}}{2}$ and their difference is $2$, it gives the conditions,

$\hookrightarrow b+c=\dfrac{a^{2}}{2} ,\ c-b=2$

By solving the equation,

$\hookrightarrow b=\dfrac{a^{2}}{4} -1,\ c=\dfrac{a^{2}}{4}+1$

$\underline{\text{Step C: Finding the triplet for given numbers.}}$

$\hookrightarrow (a,b,c)=(a,\dfrac{a^{2}-1}{2} ,\dfrac{a^{2}+1}{2} )$ if $a$ is odd.

$\hookrightarrow (a,b,c)=(a,\dfrac{a^{2}}{4} -1,\dfrac{a^{2}}{4} +1)$ if $a$ is even.

Where $a=2$,

$\hookrightarrow (a,b,c)=(2,1,3)\ \text{[Incorrect option.]}$

Where $a=3$,

$\hookrightarrow (a,b,c)=(3,4,5)\ \text{[Correct option.]}$

Where $a=6$,

$\hookrightarrow (a,b,c)=(6,17,19)\ \text{[Incorrect option.]}$

Where $a=10$,

$\hookrightarrow (a,b,c)=(10,24,26)\ \text{[Correct option.]}$

$\large{\text{\underline{Answer:-}}}$

The correct options are $(3,4,5)$ and $(10,24,26)$.