Prove that, 6 + 2 is irrational.

Let us assume that 6 + 2 is a rational number. Thus, 6 + 2 = p q ,where p and q are coprime numbers and q ≠ 0 We can write, 2 = p q - 6 2 = p - 6 q q ⇒ p - 6 q q is a rational number. But we know that 2 is an irrational number, it is contradicts to our assumption. Hence, 6 + 2 is irrational number.

Prove that, 6 + 2 is irrational.

Let us assume that 6 + 2 is a rational number. Thus, 6 + 2 = p q ,where p and q are coprime numbers and q ≠ 0 We can write, 2 = p q - 6 2 = p - 6 q q ⇒ p - 6 q q is a rational number. But we know that 2 is an irrational number, it is contradicts to our assumption. Hence, 6 + 2 is irrational number.

Prove that 7 5 is irrational.

Given, 7 5 Let us assume that 7 5 is a rational number. Then, there exists positive co-primes a , b such that, 7 5 = a b ⇒ 7 5 b = a Squaring both sides, ⇒ 245 b 2 = a 2 . . . . ( i ) Here, a 2 is a multiple of 245 . So, a should also be a multiple of 245 . ⇒ a = 245 c , where c is the other factor of a . Putting this value of a in ( i ) , ⇒ 245 b 2 = ( 245 c ) 2 ⇒ 245 b 2 = 245 2 c 2 ⇒ b 2 = 245 c 2 Here, b 2 is a multiple of 245 . So, b should also be a multiple of 245 . Here, a and b are the multiples of 245 , which contradicts our assumption that a and b are co-prime. Hence, 7 5 is irrational.

Show that any positive even integer is of the form 4 q or 4 q + 2 , where q is some integer.

Let a be any positive integer and b = 4 . Applying division lemma on a and b , a = 4 q + r , where 0 < r ≤ 4 . When r = 0 , a = 4 q + 0 = 2 ( 2 q ) When r = 1 , a = 4 q + 1 When r = 2 , a = 4 q + 2 = 2 ( 2 q + 1 ) When r = 3 , a = 4 q + 3 Since, 4 q , 4 q + 2 are multiples of 2 , these are even integers. So, every odd positive integers are in the form of either 4 q or ( 4 q + 2 ) , where q is some integer. Hence, proved.

Prove that, 6 + 2 is irrational.

Let us assume that 6 + 2 is a rational number. Thus, 6 + 2 = p q ,where p and q are coprime numbers and q ≠ 0 We can write, 2 = p q - 6 2 = p - 6 q q ⇒ p - 6 q q is a rational number. But we know that 2 is an irrational number, it is contradicts to our assumption. Hence, 6 + 2 is irrational number.

Prove that, 6 + 2 is irrational.

Let us assume that 6 + 2 is a rational number. Thus, 6 + 2 = p q ,where p and q are coprime numbers and q ≠ 0 We can write, 2 = p q - 6 2 = p - 6 q q ⇒ p - 6 q q is a rational number. But we know that 2 is an irrational number, it is contradicts to our assumption. Hence, 6 + 2 is irrational number.