Factor Me!

Assume a=2007, b=2006

And then our equation becomes:

$a\times 10001b-b\times 10001a\\ =10001ab-10001ab\\ =0$

$2007 \times 20062006 - 2006 \times 20072007$

$= (2006+1) \times 20062006 - 2006 \times (20062006 + 10001)$

$= 2006 \times 20062006 + 20062006 - 2006 \times 20062006 - 20062006$

$=\boxed{0}$

2007 x 20062006 - 2006 x 20072007

= 2007 x 2006 x 10001 - 2006 x 2007 x 10001

= 0

(2007x20062006) - (2006x20072007) = (2006 + 1) (20062006) - 2006 (20062006 +10001) =2006 x 20062006 + 20062006 - 2006 x 20062006 - 20062006 = 0.

2007 x 20062006 - 2006 x 20072007

= (2007 x 2006 x 10001) -( 2006 x 2007 x 10001)

= 0

You may check for any errors, but this is how I perceived this would be solved.

2007=2.007×10^3 ,,20062006=2.0062006×10^7 ,, 2006=2.006×10^3 ,,20072007=2.0072007×10^7,,so formula=2.007×10^3×2.0062006×10^7-2.006×10^3×2.0072007×10^7=10^10×(2.007×2.0062006-2.006×2.0072007),=10^10×(2.007×~2.006-2.006×~2.007)=0×10^10=0####

( 2007 * 20062006 ) - ( 20072007 * 2006 )=

( 2007 * 10001 * 2006 )- ( 2007 * 10001 * 2006 )=

2007 * 10001 * 2006 ( 1 - 1 )=

[ TAKING 2007 * 10001 * 2006 AS COMMON ]

2007 * 10001 * 2006 * 0=

0

$\begin{aligned} &\phantom{=}2007×20062006-2006×20072007\\ &=2007×2006×10001-2006×2007×10001\\ &=\boxed{0} \end{aligned}$

Solve: $(2007\times 20062006) - (2006\times 20072007) = x$

Multiply the expression by $\frac{2006\times 2007}{2006\times 2007}$ i.e. just multiply by $1$ , so nothing changes.

$\frac{2006\times 2007}{2006\times 2007}\left((2007\times 20062006) - (2006\times 20072007)\right) = x$

$2007$ in the denominator cancels:

$\frac{2006\times 2007}{2006}\left((20062006) - (2006\times 10001)\right) = x$

$2006$ in the denominator also cancels:

$\frac{2006\times 2007}{1}\left((10001) - (10001)\right) = x$

So we're left with:

$2006\times 2007(10001 - 10001) = x$

The bit in parenthesis is $0$ , so:

$2006\times 2007(0) = x$

$0 = x$

$\begin{aligned} (2007 × 20062006) - (2006 × 20072007) & = (2007 ×(20072007 - 10001)) - ((2007 - 1)×20072007)\\ & = (2007×20072007 - 2007×10001) - (2007×20072007 - 2007)\\ & = (2007×(20072007-1)) - (2007×(20072007 -1))\\ & = (2007×20072006) - (2007×20072006) \end{aligned}$

2007 x 20062006 = 2007 x 2006 x 10001 2006 x 20072007 = 2006 x 2007 x 10001

Therefore, the answer is 0

(2007)(20062006) - (2006)(20072007)

=(2007)(20072007-10001) - (2006(20072007)

=(2007)(20072007) - (2007)(10001) - (2006)(20072007)

=20072007 - (2007)(10001)

=20072007 - (2007)(10000+1)

=20072007 - 20072007

=0

(2007 * 2006)-(2006 * 2007) due to commutative the answer is 0

2007 x 20062006 - 2006 x 20072007
=2007 x (10001 x 2006) - 2006 x (10001 x 2007) = 0

( 2007 × 20062006 ) - ( 2006 × 20072007 ) = ( 2007 × 20062006 ) - 2006 ( 20062006 + 10001 ) = ( 2007 × 20062006 ) - ( 2006 × 20062006 ) - ( 2006 × 10001 ) = 20062006 ( 2007 - 2006 ) - ( 2006 × 10001 ) = 20062006 ( 1 ) - ( 20062006 ) = 0

simplify it to be

$(2007\times10001\times2006)-(2006\times10001\times2007)$

$=0$

A very interesting observation is that when 101, containing one cipher between two 1's, is multiplied by a two-digit number, we obtain the latter repeated as xx to xxxx. Similarly, xxx * 1001 = xxxx. So, xxxx * 10001 = xxxxxxxx.

This means, 20062006 = 2006 * 10001 20071007 = 2007 * 10001.

Thus, the answer is Cipher. :D

Same way Tuan

basically it's this:
A = 2007 and B = 2006 ; 20062006 = 2006 * 10001 20072007 = 2007 * 10001;

Therefore ( the equation is thus ) (A * 10001 B) - ( B 10001 A ) factor out the 10001 and you are left with ( a b ) - (b*a ) ==> 0

It works starting with 1 digit numbers ..
A= 7, B = 9 7 99 - 99 7 = 0 (A 11 B) - ( B 11 A)
AB - BA ==> 0

(2007x20062006)-(2006x20072007)=(2007x2006+2007x2006x10^4)-(2006x2007+2006x2007x10^4)=0

if x=2007 and y=2006

then x(10001y)-y(10001x) = 10001xy - 10001xy = 0 (Ans)

2007 x 20062006 - 2006 x 20072007.

= 40264446042 - 40264446042

= 0