Understanding HCF and LCM of Rational Numbers

The question asks about the relationship between two rational numbers if their Highest Common Factor (HCF) and Least Common Multiple (LCM) are equal. Let the two rational numbers be $a$ and $b$. We need to find out what kind of numbers $a$ and $b$ must be under the condition that $\text{HCF}(a, b) = \text{LCM}(a, b)$.

There is a fundamental property relating the HCF, LCM, and the product of two positive rational numbers. This property states:

$$ a \times b = \text{HCF}(a, b) \times \text{LCM}(a, b) $$

This is a very important concept in understanding the number properties of HCF and LCM.

Applying the Condition HCF = LCM

Given that the HCF and LCM of the two rational numbers are equal, let's say $\text{HCF}(a, b) = \text{LCM}(a, b) = k$. Note that for positive rational numbers, HCF and LCM are also positive, so $k > 0$.

Substituting this condition into the fundamental property:

$$ a \times b = k \times k $$

$$ a \times b = k^2 $$

Now, let's consider the definitions of HCF and LCM for rational numbers, particularly positive ones:

• The HCF of two positive rational numbers $a$ and $b$ is the largest rational number that divides both $a$ and $b$. This means $\text{HCF}(a, b) \le a$ and $\text{HCF}(a, b) \le b$. So, $k \le a$ and $k \le b$.
• The LCM of two positive rational numbers $a$ and $b$ is the smallest rational number that is a multiple of both $a$ and $b$. This means $\text{LCM}(a, b) \ge a$ and $\text{LCM}(a, b) \ge b$. So, $k \ge a$ and $k \ge b$.

Why the Numbers Must Be Equal

From the properties above, we have two conditions involving $k$, $a$, and $b$:

• $k \le a$ and $k \ge a$
• $k \le b$ and $k \ge b$

For $k$ to be both less than or equal to $a$ and greater than or equal to $a$ simultaneously, the only possibility is that $k = a$.

Similarly, for $k$ to be both less than or equal to $b$ and greater than or equal to $b$ simultaneously, the only possibility is that $k = b$.

Since $k = a$ and $k = b$, it logically follows that $a = b$. Therefore, if the HCF and LCM of two positive rational numbers are equal, the numbers themselves must be equal.

Let's check this with an example. If $a=4$ and $b=4$, both are rational numbers. $\text{HCF}(4, 4) = 4$ and $\text{LCM}(4, 4) = 4$. Here, HCF = LCM, and the numbers are equal.

Examining Other Options

Let's quickly consider why the other options are typically not the case when HCF and LCM are equal (unless the numbers happen to be equal in these specific cases):

• Prime Numbers: If $a$ and $b$ are two distinct prime numbers (e.g., 3 and 5), $\text{HCF}(3, 5) = 1$ and $\text{LCM}(3, 5) = 15$. HCF and LCM are not equal. If $a=b$ and they are prime (e.g., 7 and 7), $\text{HCF}(7, 7)=7$ and $\text{LCM}(7, 7)=7$. They are equal, but this requires the numbers to be equal.
• Co-prime Numbers: If $a$ and $b$ are co-prime (their HCF is 1, e.g., 4 and 9), $\text{HCF}(4, 9) = 1$ and $\text{LCM}(4, 9) = 36$. HCF and LCM are not equal. The only way for HCF = LCM = 1 is if the numbers are both 1 (for positive rational numbers), in which case they are equal.
• Compound Numbers: These are non-prime integers greater than 1. If $a$ and $b$ are distinct compound numbers (e.g., 6 and 8), $\text{HCF}(6, 8) = 2$ and $\text{LCM}(6, 8) = 24$. HCF and LCM are not equal. Again, if the compound numbers are equal (e.g., 10 and 10), $\text{HCF}(10, 10)=10$ and $\text{LCM}(10, 10)=10$. They are equal, but the numbers are equal.

In all these cases, the condition $\text{HCF}(a, b) = \text{LCM}(a, b)$ forces the numbers $a$ and $b$ to be equal (assuming positive rational numbers).

Therefore, based on the fundamental number properties of HCF and LCM, if the HCF and LCM of two rational numbers are equal, the numbers must be equal.