Let x and y be two distinct prime numbers and p = x²y³, q = xy^4, r = x^5y². Find the HCF and LCM of p, q and r. Further check if HCF (p, q, r) × LCM (p, q, r) = p × q × r or not.

HCF = xy², LCM = x^5y^4; HCF × LCM = x^6y^6 ≠ x^8y^9 = pqr

MathsCBSE Class 10 · Standard · Board exam 2025, set 30/4/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2025: question 26(B)

Set in the CBSE Class 10 board exam of 2025, paper 30/4/1, section C, question 26(B), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q26(B) · 3 marks · Short answerReal Numbers

Let $x$ and $y$ be two distinct prime numbers and $p = x^{2} y^{3}$ , $q = x y^{4}$ , $r = x^{5} y^{2}$ . Find the HCF and LCM of $p$ , $q$ and $r$ . Further check if $HCF (p, q, r) \times LCM (p, q, r) = p \times q \times r$ or not.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/4/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

For HCF take the lowest power of each prime; for LCM the highest.

Hint 2

Multiply HCF and LCM and compare with $p \times q \times r$ by adding exponents.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

HCF = product of smallest powers of common primes: $HCF (p, q, r) = x^{1} y^{2} = x y^{2}$
1 mark
LCM = product of greatest powers: $LCM (p, q, r) = x^{5} y^{4}$
1 mark
$HCF \times LCM = x y^{2} \times x^{5} y^{4} = x^{6} y^{6}$ , while $p \times q \times r = x^{2 + 1 + 5} y^{3 + 4 + 2} = x^{8} y^{9}$ .
Since $x^{6} y^{6} \neq = x^{8} y^{9}$ , $HCF (p, q, r) \times LCM (p, q, r) \neq = p \times q \times r$ .
1 mark

Answer: HCF $= x y^{2}$ , LCM $= x^{5} y^{4}$ ; HCF × LCM $= x^{6} y^{6} \neq = x^{8} y^{9} = pq r$

Where the marks are

HCF $= x y^{2}$ (1 mark)
LCM $= x^{5} y^{4}$ (1 mark)
Compares $x^{6} y^{6}$ with $x^{8} y^{9}$ and concludes they are not equal (1 mark)

Revise the chapter

Chapter: Real Numbers (notes, formulas, where marks are lost and every past question)
Topic: HCF and LCM by Prime Factorisation

Practise this chapterBoard-style questions from Real Numbers, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 30/4/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2025: question 26(B)

Hints

Step-by-step solution

Revise the chapter

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