I. HCF of p and (p + q) is G
II. HCF of p, (p - q) is G
Select the answer using the code given below :
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.
Similar Questions
- If HCF of 768 and \(x^6y^2\) is \(32xy\) for natural numbers \(x \ge 2, y \ge 2\), then what is the value of \((x + y)\)?
- What is the HCF of \(2^{36}-1\) and \(2^{45}-1\)?
The HCF of \(x\) and \(y\) is \(H\). Consider the following statements in respect of the HCF of \(p=\frac{x^3 + y^3}{x^2-xy+y^2}\) and \(q=\frac{x^3-y^3}{x^2+xy+y^2}\) :
I. The HCF of \(p\) and \(q\) can be \(H\).
II. The HCF of \(p\) and \(q\) can be \(2H\).
Which of the statements given above is/are correct?
- What is the HCF of \(x^3 + y^3 + 3xy-1\) and \((x + y)^2 - 1\)?
- A number N is such that when divided by 4, 6, 7 or 9, it leaves 3 as remainder. What is the smallest 4-digit number that satisfies this property?
- A plank of wood 4·25 m long and 3·4 m wide is to be cut into square pieces of equal size. How many square pieces of largest size can be cut from the plank, if no wastage is allowed ?
- What is the HCF of \(x^4 - 13x^2y^2 - 300y^4\), \(x^3 - 4x^2y - 4xy^2 - 5y^3\) and \(x^3 - 125y^3\)?
- Let \(N\) be a 5-digit number. When \(N\) is divided by 6, 12, 15, 24 it leaves respectively 2, 8, 11, 20 as remainders. What is the greatest value of \(N\)?
Let \(N\) be the least positive multiple of 11 that leaves a remainder of 5 when divided by 6, 12, 15, 18. Which one of he following is correct?
Let \(n\) be a natural number. The HCF of \(n, n + 10\) is 10. If the LCM is \(x\) (a 2-digit number), then how many values of \(x\) are possible?
Important Questions from LCM and HCF
The HCF and LCM of two numbers are 12 and 72, respectively. If the ratio of the two numbers is 2 ∶ 3, then the larger of the two numbers is:
Find the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case.
Joseph visits the club on every 5 th day, Harsh visits on every 24 th day, while Sumit visits on every 9 th day. If all three of them met at the club on a Sunday, then on which day will all three of them meet again?
What is the least number which when divided by 12,20 and 24 leaves in each case a remainder of 8?
Which of the following is a pair of co-primes?
