Sum of two numbers $x, y = 1050$. What is the maximum value of the HCF between $x$ and $y$?

Question

Let $HCF (x, y) = h$ then we can say that $x=ha$ and $y=hb$ for some multiple of $h$ i.e. $a$ and $b$ respectively and where $ HCF(a,b)=1$ .

As per question :-
$x+y = 1050$
$\Rightarrow ha+hb = 1050$
$\Rightarrow h(a+b) = 1050$

Now taking $h$ as maximum i.e. $1050$, then $a+b =1$ which means that either $a=0$ and $b=1$ or $a=1$ and $b=0$ and we can also see that that $HCF(a,b) = 1$ in both the cases which is valid. Then shouldn't the answer for the maximum HCF be 1050 ? Need clarification with this one please.

Thanks in advance !

Answer 1

Yes, the answer is $1050$, supposing both x and y are non-negative

If we consider $x>y$

$HCF(x;y)\le x$

So if we choose for $x$ any number less than $1050$, the $HCF$ will be strictly less than $1050$, and if we take any number greater than $1050$, $y$ would be negative.