Question

Find the HCF (Highest common factor) of the given numbers $15{{a}^{2}}{{b}^{2}}$ and $-24ab$.

Answer 1

Hint: We start solving the problem by recalling the definition and calculation of the HCF (Highest common factor) of any given numbers. We then factorize the number $15{{a}^{2}}{{b}^{2}}$ till the last possible prime number occurs. We also factorize the number $-24ab$ till the last possible prime number occurs. After factorizing the two given numbers we then multiply the common factors to get the common factor.

Complete step-by-step answer:
According to the problem, we need to find the HCF (Highest common factor) of the given numbers $15{{a}^{2}}{{b}^{2}}$ and $-24ab$.
We know that HCF (Highest common factor) is the greatest number that can divide all the given numbers. We need to factorize the given numbers first and then multiply all the common factors that were present between those two numbers.
Let us first factorize $15{{a}^{2}}{{b}^{2}}$.
We know that 15 can be written as $3\times 5$. We know that the numbers 3 and 5 are prime and cannot be factorized further.
Now ${{a}^{2}}$ can be written as $a\times a$. Since a is variable and we don’t know the absolute value of it, we can’t factorize further.
Now ${{b}^{2}}$ can be written as $b\times b$. Since a is variable and we don’t know the absolute value of it, we can’t factorize further.
So, factorization of $15{{a}^{2}}{{b}^{2}}$ is written as $15{{a}^{2}}{{b}^{2}}=3\times 5\times a\times a\times b\times b$ ---(1).
Now, we factorize $-24ab$.
We know that 24 can be written as $3\times 8$. We know that the number 3 is prime and cannot be factorized further. But, 8 is not prime and can be factorized further. We can factorize 8 as $2\times 2\times 2$. So, 24 can be factorized as $3\times 2\times 2\times 2$.
Since a and b are variables and we don’t know the absolute values of them, we can’t factorize further.
So, factorization of $-24ab$ is $-1\times 3\times 2\times 2\times 2\times a\times b$ ---(2). (Here –1 is multiplied in order to get equality on both sides).
From equations (1) and (2), we can see that the common factors are ‘3’, ‘a’ and ‘b’. But we need the highest common factor, so we multiply all the common factors to get it.
The highest common factor is $3\times a\times b=3ab$.
We have found the HCF (Highest common factor) as $3ab$.
∴ The HCF (Highest common factor) of $15{{a}^{2}}{{b}^{2}}$ and $-24ab$ is $3ab$.

Note: We need to make sure that the obtained HCF (Highest common factor) is positive as positive number is always greater than a negative number. Even if we have common negative numbers, we take the positive part and leave the negative part. We should not assume any value randomly for the variables in the numbers while factorizing. Similarly, we can expect problems that contain more variables, which may have common negative factors.