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If $${p^n}{p^m}{p^l}$$ and $${p^n}{p^m}$$, where $p$is the prime number and $n,m,l$ are integers. Then find the HCF.

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Last updated date: 24th Jul 2024
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Answer
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Hint: Here we have to find the Highest common factor (HCF) of given two numbers $${p^n}{p^m}{p^l}$$ and $${p^n}{p^m}$$ . The largest common factor of all the numbers is known as the Highest common factor of the numbers. It is also known as the Greatest Common Divisor (GCD). Here the given numbers are already in prime factorization form hence the HCF of these two numbers will be the product of the common factors present in both numbers.

Complete step-by-step answer:
The prime factors of a number are all the prime numbers with integer powers that, when multiplied together, equal the original number. We can find the prime factorization of a number by using a factor tree and dividing the number into smaller parts. The given two numbers are $${p^n}{p^m}{p^l}$$and $${p^n}{p^m}$$, where $p$is the prime number and $n,m,l$ are integers. Therefore the given numbers are in prime factorization form. So we can easily write them in factor form as,
$$\eqalign{
  & {p^n}{p^m}{p^l} = {p^n} \times {p^m} \times {p^l} \cr
  & {p^n}{p^m} = {p^n} \times {p^m} \cr} $$
The common factors present in both the numbers are $${p^n} \times {p^m}$$. Therefore the HCF of $${p^n}{p^m}{p^l}$$ and $${p^n}{p^m}$$ is $${p^n}{p^m}$$.
Another approach we can apply here is, if in the given two numbers one number is the multiple of other or in other words if one number divides the other completely then the devisor is the HCF of those two numbers. In this case $${p^n}{p^m}$$ divides $${p^n}{p^m}{p^l}$$ completely, therefore the HCF of $${p^n}{p^m}{p^l}$$ and $${p^n}{p^m}$$ is $${p^n}{p^m}$$.
So, the correct answer is “$${p^n}{p^m}$$”.

Note: Since the numbers were given in prime factor form it was easy to find their HCF. If the numbers are not in prime factor form then we can first convert them into prime factor form then we can find the HCF. Or else we can directly find the factors which divide the given numbers and then select the highest common factor present in both numbers as their HCF.