Is 7 × 5 × 3 × 2 + 3 a composite number? Justify your answer.

Composite numbers have factors other than 1 and itself. We have, 7 × 5 × 3 × 2 + 3 = 3 7 × 5 × 2 + 1 = 3 × 70 + 1 = 3 × 71 = 3 × 71 The given expression has 3 and 71 as its factors. ∴ It is a composite number.

Prove that 2 3 + 5 is an irrational number. Also check whether ( 2 3 + 5 ) ( 2 3 - 5 ) is rational or irrational.

Let 2 3 + 5 be rational equal to a b , where a and b are integers and a ≠ 0 and b ≠ 0 . Then, 2 3 + 5 = a b ⇒ 5 = a b - 2 3 ⇒ ( 5 ) 2 = a b - 2 3 2 [Squaring both sides] ⇒ 5 = a 2 b 2 - 4 a 3 b + 12 ⇒ - 7 = a 2 b 2 - 4 a 3 b ⇒ 4 a 3 b = 7 + a 2 b 2 ⇒ a 2 + 7 b 2 b 2 = 4 3 a b ⇒ 3 = a 2 + 7 b 2 4 a b Since, a and b are integers. So, a 2 + 7 b 2 4 a b is rational. Therefore, 3 is rational. Now, this contradicts the fact that 3 is irrational. Thus, our assumption was wrong. Hence, 2 3 + 5 is irrational. Now, 2 3 + 5 2 3 - 5 = 2 3 2 - 5 2 ∵ a + b a - b = a 2 - b 2 = 2 2 × 3 2 - 5 = 4 × 3 - 5 = 12 - 5 = 7 7 is rational. Therefore, 2 3 + 5 2 3 - 5 is a rational number.