Question

What is the best way to simplify
2103/2127

Answer 1

\[\dfrac{2103}{2127}\]

Answer 2

simplified is 701/709 by finding out the common factor which is 3

Answer 3

Nice! how do we know 701 and 709 have no common factors ?

Answer 4

Prime numbers

Answer 5

Is it easy to figure that out

Answer 6

We can write it as 1-8/709

Answer 7

So no common factors !

Answer 8

Interesting... could you explain how you conclude no common factors from that ?

Answer 9

Since 8 and 709 don't have any factors in common, i concluded that.

Answer 10

Like say we have this
16/4 as the number .
If i write it to be 12+4 /4 then i get 1+12/4
Here 4 and 12 have common thing in them .So we can accordingly get the factors.

Answer 11

Here's a good website describing the rules of divisibility. http://usablealgebra.landmark.edu/algebra/factoring/divisibility-rules.php

Answer 12

i agree with @samigupta8

consider the mixed fraction
\[1-\frac{a}{b}\]

such that a and b have no common factors.

then \[1-\frac{a}{b}= \frac{b-a}{b}\]

let if (b-a) and (a) have common factor (k),
we can rewrite as
\[\frac{b-a}{b}=\frac{nk}{mk} \\or\\b-a=nk\\b=mk\]

subtract the above two equaitons
we get

a=(m-n)k

but we also have b=mk

so a and b have common factor 'k'

which is contradiction with our initial suppositon that a and b have no common factors

Answer 13

Awesome! really a great observation and a cute proof !

Answer 14

yay :)

Answer 15

Loks you have derived the euclidean algorithm
\[\gcd(b,~a) = \gcd(b, ~b-a)\]

Answer 16

@baru @samigupta8