CDS 2026 (I) Elementary Mathematics, question 22

Topic: HCF and LCM

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Q22

If the HCF of pp and qq (p>q)(p>q) is GG, then which of the following statements is/are correct ?

I. HCF of pp and (p+q)(p+q) is GG

II. HCF of pp, (p−q)(p-q) is GG

Select the answer using the code given below :

  1. (a) I only
  2. (b) II only
  3. (c) Both I and IICorrect answer
  4. (d) Neither I nor II

Solution.

Answer: (c) Both I and II

  1. A common divisor of p and q also divides p+qp + q and p−qp - q.
  2. Conversely, a common divisor of p and either p+qp + q or p−qp - q must divide q by subtraction.
  3. Thus gcd⁡(p,p+q)=gcd⁡(p,p−q)=gcd⁡(p,q)=G\gcd(p,p + q) = \gcd(p,p - q) = \gcd(p,q) = G.