CBSE2026Class Class 10 · Mathematics1 Marks · Long✅ Verified
<b>Assertion (A) :</b> For any two natural numbers $a$ and $b$, the HCF of $a$ and $b$ is a factor of the LCM of $a$ and $b$.
<b>Reason (R) :</b> HCF of any two natural numbers divides both the numbers.
Directions: Select the correct answer from the codes given below —
(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.
✅ Answer & Solution
Step 1: Examine the Assertion. We know $\text{HCF}(a,b)\times \text{LCM}(a,b)=a\times b$, so
$$\text{LCM}(a,b)=\frac{a\times b}{\text{HCF}(a,b)}$$
Also HCF divides $a$, and $a$ divides LCM, hence HCF divides LCM. So Assertion (A) is TRUE.
Step 2: Examine the Reason. By definition, the HCF (highest common factor) of two numbers is a common factor, so it divides both numbers. Reason (R) is TRUE.
Step 3: Is (R) the correct explanation of (A)? Knowing only that HCF divides $a$ and $b$ is not enough — we additionally need the fact that each of $a$ and $b$ divides the LCM. So (R) alone does not fully explain (A).
Step 4: Both statements are true but (R) is not the correct explanation of (A).
Correct option is (B).
