Question

Find the HCF of 52 and 117 and express it in the form of $52x+117y$ .

Answer 1

Hint: To determine the HCF of the numbers, express the number in terms of the product of its prime factors and multiply all the common prime factors. This is the method of prime factorization. Once you get the HCF equate it with $52x+117y$ and find the integer values of x and y such that the equation is satisfied.

Complete step-by-step answer:
Before proceeding with the solution, let’s understand the concept of prime factorization. A prime number is a number which is not divisible by any other number except 1 and itself. Any number can be expressed as a product of prime numbers. All the prime numbers, which when multiplied, give a product equal to a number (say x) are called the prime factors of the number x. To write the prime factors of a number, we should always start with the smallest prime number, i.e. 2 and check divisibility. If the number is divisible by the prime number, then we write the number as a product of the prime number and another number, which will be the quotient when the given number is divided by the prime number. Then, we take the quotient and repeat the same process. This process is repeated till we are left with 1 as the quotient.
For example: Consider the number 51. It is an even number. So, it is not divisible by 2. The sum of the digits of 51 is 5 + 1 = 6. Hence, 51 is divisible by 3. Now, $51=3\times 17$ . Now, we take 17. We know, 17 is a prime number. Hence, the prime factors of 51 are 3 and 17.
Now let us find the prime factors of 52 and 117.
$\begin{align}
  & \text{ }2\left| \!{\underline {\,
  52 \,}} \right. \\
 & \text{ 2}\left| \!{\underline {\,
  26 \,}} \right. \\
 & \text{13}\left| \!{\underline {\,
  13 \,}} \right. \\
 & \text{ 01} \\
\end{align}$
$\begin{align}
  & \text{ 3}\left| \!{\underline {\,
  117 \,}} \right. \\
 & \text{ 3}\left| \!{\underline {\,
  039 \,}} \right. \\
 & \text{13}\left| \!{\underline {\,
  013 \,}} \right. \\
 & \text{ 001} \\
\end{align}$
$\begin{align}
  & 52=2\times 2\times 13 \\
 & 117=3\times 3\times 13 \\
\end{align}$
Now to find the HCF, we need to multiply all the common prime factors. As we can see, the only common factor among the two is 13. Hence we can say that:
$HCF(52,117)=13$
Now let us move to the next part of the question. We know that 117 can be written as the sum of 104 and 13.
$\therefore 117=104+13$
We know 104 is two times 52. If we use this in our equation, we get
$117\times 1=52\times 2+13$
$\Rightarrow 117\times 1-52\times 2=13$
$\Rightarrow 117\times 1+52\times (-2)=13$
If we compare this equation with the expression given in the question equated with HCF, we get
$52x+117y=13$
y=1, x=-2.
Hence, we can conclude that the HCF is 13 and x=-2 and y=1.

Note:Be careful while finding the prime factors of each number. Also, it is prescribed that you learn the division method of finding the HCF as well, as it might be helpful. If in case you are asked to find the HCF of two fractions you must use the formula $HCF=\dfrac{\text{HCF}\text{ of numerator of the fractions}}{\text{LCM of the denominator of the fractions}}$ .