Board papers: 2023–2026 · Official sample sessions: 2022–23 to 2025–26. Basic and Standard practice with NCERT-style explanations.
HCF, LCM and prime factors Key results For two positive integers, HCF × LCM = their product. The HCF divides the LCM. For distinct primes p and q, HCF = 1 and LCM = pq. A prime dividing a² also divides a. The word “prime” is essential. Question 1 · Basic Assertion (A): If the HCF of two numbers is 16 and their product is 3072, then their LCM is 162.
Reason (R): If a and b are two positive integers, then HCF × LCM = a × b.
CBSE 2025 supplementary examination · Basic · accessible paper · 430(B)/S · Q19
Answer choices for question 1 A. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C. Assertion (A) is true, but Reason (R) is false. D. Assertion (A) is false, but Reason (R) is true.
Show Answer & Explanation Official CBSE answer · Explanation by Team TutorVertex
Assertion: False. LCM = 3072 16
Reason: True. For two positive integers, HCF × LCM equals their product.
Link: The Reason gives LCM = 192 and disproves the Assertion. Answer: D.
Question 2 · Basic Assertion (A): If the HCF of 510 and 92 is 2, then their LCM is 32460.
Reason (R): HCF(a, b) × LCM(a, b) = a × b.
CBSE 2022–23 official sample · Basic · Q19
Answer choices for question 2 A. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C. Assertion (A) is true, but Reason (R) is false. D. Assertion (A) is false, but Reason (R) is true.
Show Answer & Explanation Official CBSE answer · Explanation by Team TutorVertex
Assertion: False. HCF(510, 92) = 2, so LCM = 510 × 92 2
Reason: True. For two positive integers, HCF × LCM equals their product.
Link: The Reason gives the correct LCM and disproves the Assertion. Answer: D.
Question 3 · Basic Assertion (A): The HCF of two numbers is 5 and their product is 150. Then their LCM is 40.
Reason (R): For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b.
CBSE 2023–24 official sample · Basic · Q20
Answer choices for question 3 A. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C. Assertion (A) is true, but Reason (R) is false. D. Assertion (A) is false, but Reason (R) is true.
Show Answer & Explanation Official CBSE answer · Explanation by Team TutorVertex
Assertion: False. LCM = 150 5
Reason: True. For two positive integers, HCF × LCM equals their product.
Link: The Reason gives LCM = 30 and disproves the Assertion. Answer: D.
Question 4 · Standard Assertion (A): If the product of two numbers is 5780 and their HCF is 17, then their LCM is 340.
Reason (R): HCF is always a factor of LCM.
CBSE 2022–23 official sample · Standard · Q19
Answer choices for question 4 A. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C. Assertion (A) is true, but Reason (R) is false. D. Assertion (A) is false, but Reason (R) is true.
Show Answer & Explanation Official CBSE answer · Explanation by Team TutorVertex
Assertion: True. LCM = 5780 17
Reason: True. The HCF of two positive integers is a factor of their LCM.
Link: This divisibility relationship does not determine the stated value 340, so R does not explain A. Answer: B.
Question 5 · Basic Assertion (A): For any two natural numbers a and b, the HCF of a and b is a factor of the LCM of a and b.
Reason (R): The HCF of any two natural numbers divides both the numbers.
CBSE 2025 main examination · Basic · 430/1/1 · Q19
Answer choices for question 5 A. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C. Assertion (A) is true, but Reason (R) is false. D. Assertion (A) is false, but Reason (R) is true.
Show Answer & Explanation Official CBSE answer · Explanation by Team TutorVertex
Assertion: True. The HCF of two positive integers divides their LCM.
Reason: True. The HCF is a factor of both integers.
Link: Let h be the HCF and L the LCM of a and b. Since h divides a and a divides L, h divides L. Thus R explains A. Answer: A.
Question 6 · Standard Assertion (A): HCF(36m², 18m) = 18m, where m is a prime number.
Reason (R): The HCF of two numbers is always less than or equal to the smaller number.
CBSE 2026 main examination · Standard · 30/5/1 · Q20
Answer choices for question 6 A. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C. Assertion (A) is true, but Reason (R) is false. D. Assertion (A) is false, but Reason (R) is true.
Show Answer & Explanation Official CBSE answer · Explanation by Team TutorVertex
Assertion: True. Since 36m² = (18m)(2m), 18m divides 36m². Hence HCF(36m², 18m) = 18m.
Reason: True. The HCF cannot exceed the smaller of the two positive integers.
Link: R gives an upper bound; it does not establish that 18m divides both numbers. Thus R does not explain the exact HCF. Answer: B.
Question 7 · Standard Assertion (A): The HCF of any two consecutive even natural numbers is always 2.
Reason (R): Even natural numbers are divisible by 2.
CBSE 2024–25 official sample · Standard · Q19
Answer choices for question 7 A. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C. Assertion (A) is true, but Reason (R) is false. D. Assertion (A) is false, but Reason (R) is true.
Show Answer & Explanation Official CBSE answer · Explanation by Team TutorVertex
Assertion: True. Write consecutive even natural numbers as 2k and 2k + 2. Every common divisor divides their difference, 2. Since 2 divides both numbers, their HCF is 2.
Reason: True. Every even natural number is divisible by 2.
Link: R establishes that 2 is a common factor, but does not establish that it is the greatest. Thus R does not explain A. Answer: B.
Question 8 · Basic Assertion (A): The prime numbers which divide 36 also divide 6.
Reason (R): Any number which divides p² also divides p.
CBSE 2025 main examination · Basic · 430/2/1 · Q19
Answer choices for question 8 A. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C. Assertion (A) is true, but Reason (R) is false. D. Assertion (A) is false, but Reason (R) is true.
Show Answer & Explanation Official CBSE answer · Explanation by Team TutorVertex
Assertion: True. Since 36 = 2² × 3² and 6 = 2 × 3, every prime divisor of 36 also divides 6.
Reason: False. For p = 2, the number 4 divides p² = 4 but does not divide p = 2. The claim about any number is too broad.
Link: A is true and R is false. Answer: C.
Question 9 · Standard Assertion (A): For any two prime numbers p and q, their HCF is 1 and LCM is p + q.
Reason (R): For any two natural numbers, HCF × LCM = product of the numbers.
CBSE 2025 main examination · Standard · 30/6/1 · Q19
Answer choices for question 9 A. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C. Assertion (A) is true, but Reason (R) is false. D. Assertion (A) is false, but Reason (R) is true.
Show Answer & Explanation Official CBSE answer · Explanation by Team TutorVertex
Assertion: False. For distinct primes p and q, HCF = 1 and LCM = pq. For example, 2 and 3 have LCM 6, whereas p + q = 5.
Reason: True. For two positive integers, HCF × LCM equals their product.
Link: A is false and R is true. Answer: D.
Question 10 · Standard Assertion (A): For two prime numbers x and y, where x < y, HCF(x, y) = x and LCM(x, y) = y.
Reason (R): HCF(x, y) ≤ LCM(x, y), where x and y are any two natural numbers.
CBSE 2025 main examination · Standard · 30/6/2 · Q19
Answer choices for question 10 A. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C. Assertion (A) is true, but Reason (R) is false. D. Assertion (A) is false, but Reason (R) is true.
Show Answer & Explanation Official CBSE answer · Explanation by Team TutorVertex
Assertion: False. Since x < y, the primes are distinct. Therefore HCF(x, y) = 1 and LCM(x, y) = xy.
Reason: True. The HCF divides the LCM for positive integers, so HCF ≤ LCM.
Link: A is false and R is true. Answer: D.
Question 11 · Standard Assertion (A): For two odd prime numbers x and y, with x ≠ y, LCM(2x, 4y) = 4xy.
Reason (R): LCM(x, y) is a multiple of HCF(x, y).
CBSE 2025 main examination · Standard · 30/6/3 · Q20
Answer choices for question 11 A. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C. Assertion (A) is true, but Reason (R) is false. D. Assertion (A) is false, but Reason (R) is true.
Show Answer & Explanation Official CBSE answer · Explanation by Team TutorVertex
Assertion: True. Since x and y are distinct odd primes, the prime factorisations of 2x and 4y are 2x and 2²y. Taking the greatest exponent of each prime gives LCM = 4xy.
Reason: True. The LCM is a multiple of the HCF.
Link: R states a general divisibility relationship but does not identify the prime factors needed to obtain 4xy. Thus R does not explain A. Answer: B.
Powers and the last digit Key results A positive integer ends in zero exactly when it is divisible by 10, so its prime factorisation must contain both 2 and 5. The Fundamental Theorem of Arithmetic gives a unique prime factorisation, apart from the order of the prime factors. Question 12 · Basic Assertion (A): 4ⁿ cannot end with the digit zero.
Reason (R): The prime factorisation of 4ⁿ is unique.
CBSE 2026 main examination · Basic · 430/4/1 · Q20
Answer choices for question 12 A. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C. Assertion (A) is true, but Reason (R) is false. D. Assertion (A) is false, but Reason (R) is true.
Show Answer & Explanation Official CBSE answer · Explanation by Team TutorVertex
Assertion: True. For a natural number n, 4ⁿ = 2²ⁿ has no prime factor 5. It is therefore not divisible by 10 and cannot end in zero.
Reason: True. Prime factorisation is unique apart from the order of the prime factors.
Link: Uniqueness alone does not identify the absence of 5 in 4ⁿ. Thus R does not explain A. Answer: B.
Question 13 · Standard Assertion (A): 4ⁿ ends with the digit 0 for some natural number n.
Reason (R): For a number x having 2 and 5 as its prime factors, xⁿ always ends with the digit 0 for every natural number n.
CBSE 2025 main examination · Standard · 30/4/1 · Q19
Answer choices for question 13 A. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C. Assertion (A) is true, but Reason (R) is false. D. Assertion (A) is false, but Reason (R) is true.
Show Answer & Explanation Official CBSE answer · Explanation by Team TutorVertex
Assertion: False. Since 4ⁿ = 2²ⁿ contains no prime factor 5, no natural-number power of 4 is divisible by 10.
Reason: True. If a positive integer x has prime factors 2 and 5, then x and every positive integral power xⁿ are divisible by 10 and end in zero.
Link: A is false and R is true. Answer: D.
Question 14 · Standard Assertion (A): The number 5ⁿ cannot end with the digit 0, where n is a natural number.
Reason (R): A number ends with 0 if its prime factorisation contains both 2 and 5.
CBSE 2025–26 official sample · Standard · Q19
Answer choices for question 14 A. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C. Assertion (A) is true, but Reason (R) is false. D. Assertion (A) is false, but Reason (R) is true.
Show Answer & Explanation Official CBSE answer · Explanation by Team TutorVertex
Assertion: True. The prime factorisation of 5ⁿ contains only 5 and no factor 2, so 5ⁿ cannot end in zero.
Reason: True. An integer ends in zero exactly when it is divisible by 10 = 2 × 5, requiring both prime factors 2 and 5.
Link: Applying R to 5ⁿ shows that the missing factor 2 prevents a final zero. Thus R explains A. Answer: A.
Question 15 · Standard Assertion (A): The number 5ⁿ cannot end with the digit 0, where n is a natural number.
Reason (R): Prime factorisation of 5 has only two factors, 1 and 5.
CBSE 2023 main examination · Standard · 30/5/1 · Q20
Answer choices for question 15 A. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C. Assertion (A) is true, but Reason (R) is false. D. Assertion (A) is false, but Reason (R) is true.
Show Answer & Explanation Official CBSE answer · Explanation by Team TutorVertex
Assertion: True. The number 5ⁿ has no prime factor 2, so it is not divisible by 10 and cannot end in zero.
Reason: False. Although 1 and 5 are the positive factors of 5, 1 is not prime. The prime factorisation of 5 is simply 5.
Link: A is true and R is false; ordinary factors must not be confused with prime factors. Answer: C.
Rational and irrational numbers Key results Adding a rational number to an irrational number gives an irrational number. The sum or product of two irrational numbers can be rational or irrational. A single counterexample disproves an “always” claim. For an irrationality proof, assume the expression is rational and derive a contradiction with a known irrational number. Question 16 · Basic Assertion (A): 2 + √2 is an irrational number.
Reason (R): The sum of a non-zero rational number and an irrational number is always an irrational number.
Appeared in: CBSE 2024 supplementary examination · Basic · 430/S/1 · Q20 · CBSE 2024 supplementary examination · Basic · 430/S/2 · Q19
The four choices and their order are identical in the selected appearances.
Answer choices for question 16 A. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C. Assertion (A) is true, but Reason (R) is false. D. Assertion (A) is false, but Reason (R) is true.
Show Answer & Explanation Official CBSE answer · Explanation by Team TutorVertex
Assertion: True. If 2 + √2 were rational, subtracting 2 would make √2 rational, a contradiction.
Reason: True. The sum of a rational number and an irrational number is irrational.
Link: Here 2 is rational and √2 is irrational, so R directly explains A. Answer: A.
Question 17 · Standard Assertion (A): √3 + √5 is an irrational number.
Reason (R): The sum of any two irrational numbers is always irrational.
Appeared in: CBSE 2026 main examination · Standard · 30/4/1 · Q20 · CBSE 2026 main examination · Standard · 30/4/2 · Q19
The four choices and their order are identical in the selected appearances.
Answer choices for question 17 A. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C. Assertion (A) is true, but Reason (R) is false. D. Assertion (A) is false, but Reason (R) is true.
Show Answer & Explanation Official CBSE answer · Explanation by Team TutorVertex
Assertion: True. Suppose r = √3 + √5 is rational. As r > 0, squaring r − √3 = √5 gives r² − 2 = 2r√3. Then √3 = r² − 2 2r
Reason: False. The sum of two irrational numbers need not be irrational: √2 + (−√2) = 0.
Link: A is true and R is false. Answer: C.
Question 18 · Basic Assertion (A): √2(5 − √2) is an irrational number.
Reason (R): The product of two irrational numbers is always irrational.
CBSE 2024 main examination · Basic · 430/3/1 · Q20
Answer choices for question 18 A. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C. Assertion (A) is true, but Reason (R) is false. D. Assertion (A) is false, but Reason (R) is true.
Show Answer & Explanation Official CBSE answer · Explanation by Team TutorVertex
Assertion: True. √2(5 − √2) = 5√2 − 2. If this were rational, say r, then √2 = r + 2 5
Reason: False. The product of two irrational numbers need not be irrational: √2 × √2 = 2.
Link: A is true and R is false. Answer: C.
Question 19 · Basic Assertion (A): (2 + √3)√3 is an irrational number.
Reason (R): The product of two irrational numbers is always irrational.
CBSE 2024–25 official sample · Basic · Q20
Answer choices for question 19 A. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C. Assertion (A) is true, but Reason (R) is false. D. Assertion (A) is false, but Reason (R) is true.
Show Answer & Explanation Official CBSE answer · Explanation by Team TutorVertex
Assertion: True. (2 + √3)√3 = 3 + 2√3. If this were rational, say r, then √3 = r − 3 2
Reason: False. The product of two irrational numbers need not be irrational: √2 × √2 = 2.
Link: A is true and R is false. Answer: C.
Question 20 · Basic Assertion (A): (a + √b)(a − √b) is a rational number, where a and b are positive integers.
Reason (R): The product of two irrational numbers is always rational.
CBSE 2025 main examination · Basic · 430/6/1 · Q19
Answer choices for question 20 A. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C. Assertion (A) is true, but Reason (R) is false. D. Assertion (A) is false, but Reason (R) is true.
Show Answer & Explanation Official CBSE answer · Explanation by Team TutorVertex
Assertion: True. Using (u + v)(u − v) = u² − v² gives (a + √b)(a − √b) = a² − b, which is an integer and hence rational.
Reason: False. The product of two irrational numbers need not be rational: √2 × (1 + √2) = 2 + √2 is irrational.
Link: A is true and R is false. Answer: C.
Question 21 · Standard Assertion (A): The perimeter of triangle ABC shown in the figure is a rational number.
Reason (R): The sum of the squares of two rational numbers is always rational.
A B C 2 cm 3 cm Official question figure redrawn by TutorVertex. CBSE 2023 main examination · Standard · 30/1/1 · Q20
Answer choices for question 21 A. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). C. Assertion (A) is true, but Reason (R) is false. D. Assertion (A) is false, but Reason (R) is true.
Show Answer & Explanation Official CBSE answer · Explanation by Team TutorVertex
Assertion: False. By Pythagoras’ theorem, AC² = 2² + 3² = 13, so AC = √13 cm. Since 13 is not a perfect square, the perimeter 5 + √13 cm is irrational.
Reason: True. The square of a rational number is rational, and the sum of two rational numbers is rational.
Link: R makes AC² rational, but does not make AC or the perimeter rational. A is false and R is true. Answer: D.
Common questions How many Assertion–Reason questions come in the CBSE Class 10 Maths board paper? The official 2026–27 Basic and Standard sample-paper format has two Assertion–Reason questions, Q19 and Q20, in Section A, worth one mark each. These are across the whole paper, not two from every chapter.
What if both the Assertion and Reason seem false? None of the four standard choices used here represents both statements being false. Recheck the exact wording, conditions and calculations. If both are still false, flag a possible question or transcription error rather than forcing an answer.
How do I choose between A and B? Both choices require the Assertion and Reason to be true. Choose A only when the Reason explains the Assertion; otherwise choose B. Follow the actual choice order printed with each question.
Can one example disprove an “always” statement? Yes. For example, √2 × √2 = 2 disproves the claim that the product of two irrational numbers is always irrational.
Does having 2 as a common factor prove that the HCF is 2? No. You must also establish that no greater common factor is possible. For consecutive even numbers, every common divisor divides their difference, 2.
Why does a number ending in zero need both 2 and 5? It is divisible by 10, and 10 = 2 × 5. Both primes must occur in its prime factorisation.
How this question bank is prepared This selected bank contains 21 distinct Assertion–Reason questions across 23 checked appearances: 17 board appearances and 6 official sample-paper appearances. Two selected repeats are merged, with their appearances and identical choice order noted. Other repeated appearances may exist.
Statements, mathematical symbols, diagrams and the four choices are checked against official question papers. Answers are checked against the corresponding marking schemes and verified mathematically. Each explanation is written by Team TutorVertex in three labelled parts: Assertion, Reason and Link, addressing the truth of both statements and their explanatory connection. No incorrect-choice explanations are included.
The current NCERT chapter and CBSE scope centre on prime factorisation and irrationality. The triangle question applies rationality to a perimeter using Pythagoras’ theorem. Its figure is redrawn with the given side lengths and right angle preserved. No calculated hypotenuse or perimeter is added to the question diagram.
Available official archives were screened with text extraction and OCR where needed. Only selected checked appearances are counted; this is not an exhaustive transcription of every paper or language edition. Ordinary MCQs and written case studies are kept separate. The 2026–27 sample papers were checked, but their Assertion–Reason questions do not supply an additional Real Numbers item for this selection. Questions with unclear printed terminology are held back for review rather than silently rewritten.