Question 7

We need to check if the number 3721 is actually a prime number.

Step 1 — Check the number 3721

Let's find the square root of 3721. We can try numbers close to its square root. We know that $60 \times 60$ is 3600. Let's try multiplying 61 by 61.

$61 \times 61$ $= 3721$

$\boxed{\sqrt{3721} = 61}$

So, $\sqrt{3721}$ is 61. This means 3721 is a perfect square. A perfect square greater than 1 is not a prime number. So, 3721 is not a prime number. It is a composite number.

Step 2 — Answer the questions

The problem gives us two statements. First, $\sqrt{p}$ is irrational for all primes $p$. Second, it asks us to suppose 3721 is a prime. Based on these two statements, we can make a conclusion.

Answer

(i) Yes, we can conclude that $\sqrt{3721}$ is an irrational number. (ii) No, our conclusion is not correct. (iii) The premise that 3721 is a prime number is false. We found that $\sqrt{3721}$ is 61. So, 3721 is 61 multiplied by 61. This means 3721 is a composite number. Since 3721 is not prime, the given condition does not apply. $\sqrt{3721}$ is 61, which is a rational number.