Question
Using Euclid's division algorithm, find whether the pair of numbers $847,2160$ are co$-$primes or not.
✓
Answer
$a=2160, b=847$By Euclid's lemma, given positive integers $a$ and $b$ , there exist unique integers $q$ and $r$ satisfying
$a=b q+r, 0 \leq r < b$.
As $2160>847$, we apply the division lemma to $2160$ and $847$ , to get $2160=847 \times 2+466$
Since the remainder $466 \neq 0$, we apply the division lemma to $847$ and $466$ , and continue the same process till we get remainder $0 .$
$847=466 \times 1+381$
$466=381 \times 1+85$
$381=85 \times 4+41$
$85=41 \times 2+3$
$41=3 \times 13+2$
$3=2 \times 1+1$
$2=1 \times 1+1$
$1=1+0$
As $1$ is the $\text{HCF}$ of $847$ and $2160.847$ and $2160$ are the co$-$primes.
$a=b q+r, 0 \leq r < b$.
As $2160>847$, we apply the division lemma to $2160$ and $847$ , to get $2160=847 \times 2+466$
Since the remainder $466 \neq 0$, we apply the division lemma to $847$ and $466$ , and continue the same process till we get remainder $0 .$
$847=466 \times 1+381$
$466=381 \times 1+85$
$381=85 \times 4+41$
$85=41 \times 2+3$
$41=3 \times 13+2$
$3=2 \times 1+1$
$2=1 \times 1+1$
$1=1+0$
As $1$ is the $\text{HCF}$ of $847$ and $2160.847$ and $2160$ are the co$-$primes.
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