Understanding HCF Properties

We are given that the Highest Common Factor (HCF) of \(p\) and \(q\) is \(G\), where \(p > q\). This means we can express \(p\) and \(q\) as:

• \(p = Gx\)
• \(q = Gy\)

Here, \(x\) and \(y\) are coprime integers (their HCF is 1), and since \(p > q\), we know \(x > y\).

Analyzing Statement I

Statement I checks the HCF of \(p\) and \((p + q)\).

Let's find \((p + q)\): \(p + q = Gx + Gy = G(x + y)\)

Now, we find the HCF of \(p\) and \((p + q)\): \(HCF(p, p + q) = HCF(Gx, G(x + y))\)

Using the property \(HCF(ka, kb) = k \times HCF(a, b)\), we get: \(HCF(p, p + q) = G \times HCF(x, x + y)\)

We know that \(HCF(a, b) = HCF(a, b - a)\). Applying this: \(HCF(x, x + y) = HCF(x, (x + y) - x) = HCF(x, y)\)

Since \(x\) and \(y\) are coprime, \(HCF(x, y) = 1\). Therefore: \(HCF(p, p + q) = G \times 1 = G\)

So, Statement I is correct.

Analyzing Statement II

Statement II checks the HCF of \(p\) and \((p - q)\).

Let's find \((p - q)\): \(p - q = Gx - Gy = G(x - y)\)

Now, we find the HCF of \(p\) and \((p - q)\): \(HCF(p, p - q) = HCF(Gx, G(x - y))\)

Applying the property \(HCF(ka, kb) = k \times HCF(a, b)\): \(HCF(p, p - q) = G \times HCF(x, x - y)\)

Using the property \(HCF(a, b) = HCF(a, a - b)\) or similar difference properties: \(HCF(x, x - y) = HCF(x - (x - y), x - y) = HCF(y, x - y)\)

Alternatively, \(HCF(x, x - y) = HCF(x, y)\) because any common factor of \(x\) and \(x-y\) must also divide their difference, which is \(y\). Since \(x\) and \(y\) are coprime, \(HCF(x, y) = 1\). Therefore: \(HCF(x, x - y) = 1\)

So, \(HCF(p, p - q) = G \times 1 = G\).

Statement II is also correct.

Conclusion

Both Statement I and Statement II are correct. Thus, the correct option includes both.