No, 72 and 20 cannot be the least common multiple (LCM) and highest common factor (HCF) of the same two numbers.
The reason for this lies in the relationship between LCM and HCF. For any two numbers, the product of their LCM and HCF is equal to the product of the two numbers themselves. Mathematically, this is expressed as:
LCM(a, b) × HCF(a, b) = a × b
In our case, if we consider 72 as the LCM and 20 as the HCF, we can set up the equation:
72 × 20 = a × b
This simplifies to:
1440 = a × b
Next, we need to find integers a and b such that their HCF is 20. The multiples of 20 can be expressed as:
a = 20m
b = 20n
where m and n are co-prime integers (the only common factor they share is 1). Substituting this into our equation gives:
1440 = (20m) × (20n) = 400mn
Dividing both sides by 400, we get:
3.6 = mn
Since m and n must be integers, and the product of two integers cannot equal a non-integer (3.6), it follows that no such integers a and b exist that meet the conditions of having 72 as the LCM and 20 as the HCF. Thus, it’s not possible for 72 and 20 to be the LCM and HCF of the same two numbers.