數學問題 20分

(1) Find the smallest positive value of x such that (4x8+3x7+2x5+x) mod 5 = 0
where x is known as the check digit of the 4-digit number 875x

(2a) Show that √3 is an irrational number [use "Proof by contradiction"]

(b) Prove the follow statement by the method of "Proof by contradicition" :
THe difference between a rational and an irrational number is irrational

(C) Find a counter example to disprove the following statement
THe difference between two different irrational number is irrational

(d) Given 12 integers, show that at least two of them can be chosen such that
their difference is divisible by 11