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

Last updated date: 18th Jun 2024
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Hint: In order to solve this problem use Euclid division lemma theorem. Then assume the number according to the theorem. You also need to know that The greatest number that is a factor of two (or more) other numbers. When we find all the factors of two or more numbers, and some factors are the same ("common"), then the largest of those common factors is the Greatest Common Factor. Abbreviated "GCF". Also called "Highest Common Factor". Knowing these things will solve this problem.

To find HCF we use the Euclid division lemma on 657 and 963.
Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. It was first proved by Euclid in his work Elements. There are several proofs of the theorem.
We know that as per the Euclid division lemma, a = bq + r, 0$\leqslant$r$\leqslant$b
Where a = dividend, b = divisor, q = quotient, r = remainder.
963 = 657 x 1 + 306
Now here, Dividend a = 963, Divisor b = 657, Quotient q = 1.
And remainder = 306.
Since remainder $\ne$ 0, then again apply Euclid division lemma on division 657 and remainder 306.
657 = 306 x 2 + 45
Since remainder$\ne 0$, then again apply Euclid division lemma on division 306 and remainder 45.
306 = 45x6 + 36
Since remainder$\ne 0$, then again apply Euclid division lemma on division 45 and remainder 36.
45 = 35 x 1 +9
Since remainder$\ne 0$, then again apply Euclid division lemma on division 36 and remainder 9.
36 = 9 x 4 + 0
Now, the remainder becomes 0 So, we stop applying the Euclid division lemma.
When the remainder becomes 0 then the HCF is the divisor of that equation.
HCF = 9
657x + 963(-15) = 9
657x = 14445 + 9
x = 22
So, the value of x is 22.

Note: Whenever we face such types of problems we use some important points. Like always use the Euclid division lemma unless the remainder becomes zero. When the remainder comes to zero then HCF is equal to the divisor of that Euclid division lemma equation. Doing this will solve your problem.