Question

If d is the HCF of 30 and 72 with a linear equation $ d=30x+72y $ find x and y.

Answer 1

Hint: We will first understand the Euclid’s Division Lemma which we are going to use here for finding Highest common factor (HCF). It is given as if two positive numbers ‘a’ and ‘b’, then there exists unique integers ‘q’ and ‘r’ such that which satisfies the condition $ a=bq+r $ where $ 0\le r\le b $ . Then we will divide 72 by 30 and will get the equation in form of $ dividend=\left( divisor\cdot quotient \right)+remainder $ . Using that equation, which we got we will substitute values in such a way that we will get one equation similar to $ d=30x+72y $ . By comparing we will get an answer of x and y.

Complete step-by-step answer:
Here, we will first understand Euclid’s Division Lemma which we will be using to find the Highest Common factor (HCF).
It states that if two positive numbers ‘a’ and ‘b’, then there exists unique integers ‘q’ and ‘r’ such that which satisfies the condition $ a=bq+r $ where $ 0\le r\le b $ . Basically, in the form of $ dividend=\left( divisor\cdot quotient \right)+remainder $ .
Also, HCF is the greatest number which exactly divides two or more positive integers.
So, here we have two numbers 30 and 72. As 72 is greater, we will divide that by 30. So, we get as
 $ 72=30\times 2+12 $ …………………….(1)
This is in the form of $ dividend=\left( divisor\cdot quotient \right)+remainder $ .
Now, we will take the divisor as dividend and remainder as divisor to solve. So, we get as
 $ 30=12\times 2+6 $ …………………….(2)
Again, we will do the same so we can write as
 $ 12=6\times 2+0 $ …………………..(3)
As the remainder is 0, we can not solve further. So, we can consider the divisor to be HCF of 30 and 72 i.e. 6.
Now, we got d to be 6 as per the question i.e. $ 6=30x+72y $ So, we have to find now x and y values.
So, from equation (2), we will make 6 as subject variable and on re-arranging the terms we get as
 $ 6=30-12\times 6 $
From equation (1), we will put a value of 12 in the above equation. So, first on finding value of 12 we get as
 $ 12=72-30\times 2 $
On substituting this value, we get equation as
 $ 6=30-\left[ \left( 72-30\times 2 \right)\times 2 \right] $
On further expanding the brackets we get as
 $ 6=30-\left( 72\times 2-30\times 2\times 2 \right) $
 $ 6=30-72\times 2+30\times 4 $
Now, if we take 30 common we get as and giving minus sign to 2 instead of 72 we get as
 $ 6=30\left( 1+4 \right)+72\times \left( -2 \right) $
 $ 6=30\left( 5 \right)+72\times \left( -2 \right) $
We will compare the above equation with our equation i.e. $ 6=30x+72y $ . We can say that $ x=5,y=-2 $ .
Thus, we got values of x, y as $ x=5,y=-2 $ .

Note: Remember that for this type of problem we cannot find HCF by using factorization method or any other method. By that we will only get HCF, but we will not be able to get that same equation of which we want to find x and y values. So, students should know when and how to use Euclid’s Division Lemma method to get the correct answer. Here, finding HCF is not necessary but we have found HCF in order to substitute values into one another and get answers.