Let H be the HCF of x and y (natural numbers). Let p=(x^3+y^3)/(x^2-xy+y^2) and q=(x^3-y^3)/(x^2+xy+y^2). Consider: I. HCF of p,q can be H. II. HCF of p,q can be 2H. Which is/are correct?

CDS 2025 PYQ with Answer — Let H be the HCF of x and y (natural numbers). Let p=(x^3+y^3)/(x^2-x…

2025 · Q12

Let H be the HCF of x and y (natural numbers). Let p=(x^3+y^3)/(x^2-xy+y^2) and q=(x^3-y^3)/(x^2+xy+y^2). Consider:

IHCF of p,q can be H.
IIHCF of p,q can be 2H. Which is/are correct?

Elementary MathematicsHCF and LCM

AI only
BII only
CBoth I and II
DNeither I nor II

Explanation

In fact p=(x^3+y^3)/(x^2-xy+y^2)=x+y and q=(x^3-y^3)/(x^2+xy+y^2)=x-y. Now gcd(x+y,x-y) equals H when x,y have opposite parity (e.g. x=3,y=2: gcd(5,1)=1=H) and equals 2H when both are odd (e.g. x=3,y=1: gcd(4,2)=2=2H). Hence the HCF can be H (statement I) and can be 2H (statement II): both are correct. (A published coaching key says A, but the 'can be' phrasing makes C correct.)