Question 6

If two positive integers $m$ and $n$ are expressible in the form $m=p{q}^3$ and $n=p^3{q}^2$ , where $p,q$ are prime numbers, then HCF $(m,n)=$

Solution:

To find the HCF (Highest Common Factor) of the two positive integers $$m$$ and $$n$$ expressed as $$m=pq3$$and $$n=p3q2$$, where $$p$$ and $$q$$ are prime numbers, we can follow these steps:

Step 1: Write down the expressions for $$m$$ and $$n$$
We have:
– $$m=pq3$$
– $$n=p3q2$$

Step 2: Factorize $$m$$ and $$n$$
From the expressions, we can see:
– For $$m$$: The prime factorization is $$p^1\cdot q^3$$
– For $$n$$: The prime factorization is $$p^3\cdot q^2$$

Step 3: Identify the common prime factors
The common prime factors between $$m$$ and $$n$$ are $$p$$ and $$q$$.

Step 4: Determine the minimum power of each common prime factor
– For $$p$$: The minimum power is $$min(1,3)=1$$
– For $$q$$: The minimum power is $$min(3,2)=2$$

Step 5: Write the HCF using the common prime factors
The HCF is obtained by multiplying the common prime factors raised to their minimum powers:
$$\text{HCF}(m,n)=p^{min(1,3)}\cdot q^{min(3,2)}=p^1\cdot q^2=p{q}^2$$

Conclusion
Thus, the HCF of $$m$$ and $$n$$ is:
$$\text{HCF}(m,n)=p{q}^2$$