If the HCF of 657 and 963 is expressible in the form $657 x+963 x-15$, find $x$.

Question:

If the HCF of 657 and 963 is expressible in the form $657 x+963 x-15$, find $x$.

Solution:

We need to find $x$ if the H.C.F of 657 and 963 is expressible in the form $657 x+963 y(-15)$.

Given integers are 657 and 963.

By applying Euclid's division lemma, we get $963=657 \times 1+306$.

Since the remainder $\neq 0$, so apply division lemma on divisor 657 and remainder 306

$657=306 \times 2+45$

Since the remainder $\neq 0$, so apply division lemma on divisor 306 and remainder 45

$306=45 \times 6+36$

Since the remainder $\neq 0$, so apply division lemma on divisor 45 and remainder 36

$45=36 \times 1+9$

Since the remainder $\neq 0$, so apply division lemma on divisor 36 and remainder 9

Therefore, H.C.F. = 9.

Given H.C.F $=657 x+936(-15)$

Therefore,

$\Rightarrow \quad 9=657 x-14445$

$\Rightarrow \quad 9+14445=657 x$

$\Rightarrow \quad 14454=657 x$

$\Rightarrow \quad x=\frac{14454}{657}$

$\Rightarrow \quad x=22$.