JAMIA 2026 — Reasoning PYQ

1. Lower Bound of LCM:

The Least Common Multiple ($x$) of two positive integers $a$ and $b$ must be a multiple of both numbers. Therefore, it cannot be smaller than either of the two numbers. It must be greater than or equal to the larger of the two numbers:

$$x\ge \max (a,b)$$

2. Upper Bound of LCM:

The highest possible value for the LCM of two numbers occurs when the numbers are co-prime (i.e., $\text{HCF}(a,b)=1$). In that case, the LCM is exactly equal to their product. For all other cases, it is strictly less than their product:

$$x\le (a\times b)$$

3. Combine the Inequalities:

Combining both bounds gives the absolute boundary range for the LCM of any two positive integers:

$$\max (a,b)\le x\le (a\times b)$$

Correct Answer

Option (a) $\max (a,b)\le x\le (a\times b)$