what are the applications of Langrange theorem in daily life? where we use it?

Which Lagrange theorem would that be?

The one in Number theory?

or

The one in Group theory?

As this question was posted in the Linear Algebra and Group Theory section it seems reasonable to assume it was referring to the theorem that the order of any group is an integer multiple of that of any of it's subgroups (proof is easy - try it; hint - use the definition of cosets)

Whether it applies in so-called daily life, I have no idea.

Should we care?

That would be applied usually for finite groups. I imagine it has a use for elliptic curve cryptography over finite fields. though in this area it is hard to tell what is used exactly where, especially anything like this, which is fairly basic.

Maybe it is worth to care, at least from the point of the educationer. After you prove it once and provide a suitable amount of exercise work,  you hope that it becomes second nature to the students.

And it may be more relevant to the "daily life" of a mathematician than, say, a plumber.

3 hours ago, Strange said:

And it may be more relevant to the "daily life" of a mathematician than, say, a plumber.

Unsurprisingly, discussion abounds, e.g.

www.intmath.com/blog/learn-math/learn-math-for-plumbing-6423

A keyword being "mathematical literacy". Maybe engaging with basic examples of abstract algebra is good training for solving various questions that come up.

Relevance to daily life seems a bit broader than what the OP has in mind. The plumber may enjoy to read literature, without necessarily having an answer ready for how "Hamlet" is applicable in a work situation. He/she may also alternatively simply enjoy to think about mathematics.

1 hour ago, taeto said:

Unsurprisingly, discussion abounds, e.g.

www.intmath.com/blog/learn-math/learn-math-for-plumbing-6423

I had a feeling someone was going to say something like that...

I suppose one application is as a hint for pattern-recognition, it can give you an idea what to look for.

An example could be when one considers symmetrical objects, this is daily life for some.. in this context the theorem (which is so natural that you never really think about it, you just use it) can hint at components of the object which can correspond to sub-symmetries.. so in some sense you loosely use it together with some implicit monotone galois connection.

Though i dont know what a plumber would do with it unless they're pulling some Good Will Hunting bit...