u/Delicious-Wrap-1441

Q= show that if (a,b)=5, then infinite a and b exists for which a+b=100

My answer= via bezout identity

a(x)+b(y)=5

Then as x,y belong to S(non empty set of real numbers)

a(20.x) +b(20.y)= 100

20.x= u

20.y=v

a(u)+b(v)=100

Hence proved.

Ik the method of taking a=100k-5, but I wanted to see how far I've come and measure my progress because I devised the solution on my own.

I was solving the question, if (a,b)=1 then prove that gcd of a^p-b^p/ a-b, a-b is either p or 1, i understand why the gcd might be one, but unable to understand why it may be p? Because,as according to the solution, you use the method of pb^p-1 congruent pa^p-1mod a-b, this got me wondering, why would I even think to use mod a-b in this situation? Which got me thinking what does remainders have to do with factors? Experts pls help

I'm broke as hell, can't afford any tutoring on music, but I really really really want to play those lovely pieces I listen to all the time

Now I'm in a fix, piano or violin? Both are costing me money for buying the actual instrument, but I've heard violin is extremely difficult to learn on your own and the unnatural posture will be bad for me if I don't have a tutor to correct it since I already have neck pain due to studying. And piano, I've heard u need atleast like 61 weighted keys for western classical. So experts, what should I do?