Can 72 and 20 be the least common multiple (LCM) and highest common factor (HCF) of two numbers? If so, why?

To determine if 72 can be the least common multiple (LCM) and 20 can be the highest common factor (HCF) of two numbers, we first need to understand the definitions and relationships between LCM and HCF.

The LCM of two numbers is the smallest number that is a multiple of both, while the HCF (or GCD – Greatest Common Divisor) is the largest number that divides both without leaving a remainder. A key property of LCM and HCF is that for two integers a and b, the following relation holds:

LCM(a, b) × HCF(a, b) = a × b

In our scenario, if we let the LCM be 72 and the HCF be 20, we apply the relationship:

72 × 20 = a × b

This results in:

1440 = a × b

Next, let’s examine the factors of 20 and ensure they are compatible with 72. The prime factorization of 20 is:

  • 20 = 22 × 5

And the prime factorization of 72 is:

  • 72 = 23 × 32

For the two numbers to have an HCF of 20, they both must share 20 as a common factor. This means the two numbers can be expressed in the form:

  • Number 1 = 20 × m
  • Number 2 = 20 × n

where m and n are integers that do not share any factors with 20 (to maintain the HCF). Therefore, their product becomes:

Number 1 × Number 2 = (20 × m) × (20 × n) = 400 × (m × n)

We now have:

1440 = 400 × (m × n)

By dividing both sides by 400, we find:

m × n = 3.6

Since m and n must be integers, this equation indicates that it is impossible for m and n to be whole numbers. Thus, there is no pair of integers whose product equals 3.6.

Consequently, we conclude that 72 and 20 cannot simultaneously be the LCM and HCF of two numbers. The mismatch between the integer requirement for m and n illustrates this inconsistency. In summary, 72 and 20 cannot serve as the LCM and HCF of two numbers due to the violation of the fundamental relationship between them.

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