Question

If the HCF of 657 and 963 is expressible in the form \[657x - (963 \times 15)\], then find x.

Answer 1

Hint: HCF of two numbers can be found by the prime factorization method. Equate the HCF of 657 and 963 to \[657x - (963 \times 15)\] and solve for x.

Complete step-by-step answer:
HCF or Highest Common Factor of two numbers is the largest number that divides both the numbers completely without leaving a remainder.
The HCF of two numbers can be found by the prime factorization method.
In this method, we first write the numbers as the product of the prime numbers.
The prime factors of 657 are 3, 3, and 73. 73 is a prime number and can’t be factorized further.
The prime factors of 963 are 3, 3, and 107. 107 is a prime number and can’t be factorized further.
\[657 = 3 \times 3 \times 73\]
\[963 = 3 \times 3 \times 107\]
Next, we need to find common factors. The common factors of 657 and 963 are 3, 3.
The HCF is the product of all common factors.
\[HCF = 3 \times 3\]
\[HCF = 9\]
We now equate \[657x - (963 \times 15)\] to 9.
\[657x - (963 \times 15) = 9\]
Simplifying, we have:
\[657x - 14445 = 9\]
Taking 14445 to the right-hand side of the equation and adding with 9, we get:
\[657x = 9 + 14445\]
\[657x = 14454\]
Solving for x, we have:
\[x = \dfrac{{14454}}{{657}}\]
We know that 14454 is obtained by 22 multiplied with 657, hence, when 14454 is divided by 657, we get 22.
\[x = 22\]
Hence, the value of x is 22.

Note: You can also use Euclid’s division to find the HCF of 657 and 963. Remember the fact that 107 and 73 are prime numbers.