Question

If HCF \[(225,60) = 225 \times 5 - 10x\], then find the value of \[x\].

Answer 1

Hint:
We will first find the factors of the given numbers 225 and 60 by using the prime factorization method. We will then find the HCF of the given numbers using the obtained factors. Then we will equate the RHS to the obtained HCF and then find the value of \[x\].

Complete step by step solution:
We are given that HCF \[(225,60) = 225 \times 5 - 10x\].
We know that HCF means the “Highest Common Factor”. We can find the HCF of two numbers by first prime factorizing them. Then, we must take the product of the common factors in their lowest powers.
Let us first prime factorize 225.
\[225 = 3 \times 3 \times 5 \times 5\]
Here, 3 and 5 are both prime numbers.
Now, let us prime factorize 60. Therefore, we get
\[60 = 2 \times 2 \times 3 \times 5\]
Here 2, 3, and 5 are all prime.
Now, we will find the HCF of 225 and 60.
Here, we observe that factors 3 and 5 are common to both 225 and 60. Also, 3 and 5 are both common with the power 1 i.e., \[{3^1}\] and \[{5^1}\] are common factors. So, their product will also be a common factor.
 HCF \[(225,60) = 3 \times 5 = 15\]
So, 15 is a common factor to both 225 and 60, and this is the highest.
Now, we have to equate the polynomial \[225 \times 5 - 10x\] to this HCF. Thus, we get
\[225 \times 5 - 10x = 15\]
Let us transpose 15 to the LHS and \[10x\] to the RHS. This gives us
\[ \Rightarrow 225 \times 5 - 15 = 10x\]
\[ \Rightarrow 1110 = 10x\]
Dividing on both sides of the above equation by 10, we get
\[ \Rightarrow \dfrac{{1110}}{{10}} = \dfrac{{10x}}{{10}}\]
\[ \Rightarrow x = 111\]

Therefore, we get the value of \[x\] as 111.

Note:
Highest common factor or H.C.F. of two numbers is the largest number that divides both the numbers and it is also known as the greatest common divisor. While finding HCF of two or more numbers, we have to consider the product of common factors in their lowest powers. On the other hand, while finding LCM of two or more numbers, we must consider the product of common factors in their highest powers.