Check the validity of the following statement.

p : 125 is divisible by 5 and 7.

Identify the Quantifiers in the following statements.

(i) There exists a triangle which is not equilateral.

(ii) For all real numbers x and y, xy = yx.

(iii) There exists a real number which is not a rational number.

(iv) For every natural number x, x + 1 is also a natural number.

(v) For all real numbers x with x > 3, x2 is greater than 9.

(vi) There exists a triangle which is not an isosceles triangle

(vii) For all negative integers x, x³ is also a negative integers.

(viii) There exists a statement in above statements which is not true.

(ix) There exists a even prime number other than 2.

(x) There exists a real number x such that x² + 1 = 0.

Prove by direct method that for any integer ‘n’, n³ – n is always even.

Check the validity of the following statement.

q : 131 is a multiple of 3 or 11.

Prove the following statement by contradication method.

p : The sum of an irrational number and a rational number is irrational.