MHB What is the Correct Value of n in this Given Ratio?

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$n=\dfrac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)}$

$find\,\,\, n$
 
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Albert said:
$n=\dfrac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)}$

$find\,\,\, n$

we have $x^4+ 324 = (x^2)^2 + 18^2 = (x^2+18)^2 - 36x^2 = (x^2 + 6x + 18)(x^2-6x+18$
using this we get
$10^4 + 324 = (10 * 16 + 18)(10 * 4 + 18)$
$22^4 + 324 = (22 * 28 + 18)(22 * 16 + 18)$
$34^4 + 324 = (34 * 40 + 18)(34 * 28 + 18)$
$46^4 + 324 = (46 * 52 + 18)(46 * 40 + 18)$
so numerator $= (10^4+324)(22^4+324)(34^4+324)(46^4+324)$
$= ( 4 * 10 + 18)(10 * 16 + 18) (16 * 22 + 18) ( 22 * 28 + 18)(28 * 34 + 18) (34 * 40 + 18) (40 * 46 + 18) (46 * 52 + 18)$
now for denominator
$4^4 + 324 = (4 * 10 + 18)( 4 * (-2) + 18)$
$16^4 + 324 = (16 * 22 + 18)( 16 * 10 + 18)$
$28^4 + 324 = (28 * 34 + 18)(28 * 22 + 18)$
$40^4 + 324 = (40 * 46 + 18)(40 * 34 + 18)$
so denominator $= (4^4+324)(16+324)(28+324)(40+324)$
$= ( 4 * (-2) + 18)(4 * 10 +18) (10 * 16 + 18) ( 16 * 22 + 18)(22 * 28 + 18) (28 * 34 + 18) (34 * 40 + 18) (40 *46 + 18)$
so ratio = $\frac{46* 52 +18}{4 * (-2) + 18} = 240$
or n = 240
 
kaliprasad said:
we have $x^4+ 324 = (x^2)^2 + 18^2 = (x^2+18)^2 - 36x^2 = (x^2 + 6x + 18)(x^2-6x+18$
using this we get
$10^4 + 324 = (10 * 16 + 18)(10 * 4 + 18)$
$22^4 + 324 = (22 * 28 + 18)(22 * 16 + 18)$
$34^4 + 324 = (34 * 40 + 18)(34 * 28 + 18)$
$46^4 + 324 = (46 * 52 + 18)(46 * 40 + 18)$
so numerator $= (10^4+324)(22^4+324)(34^4+324)(46^4+324)$
$= ( 4 * 10 + 18)(10 * 16 + 18) (16 * 22 + 18) ( 22 * 28 + 18)(28 * 34 + 18) (34 * 40 + 18) (40 * 46 + 18) (46 * 52 + 18)$
now for denominator
$4^4 + 324 = (4 * 10 + 18)( 4 * (-2) + 18)$
$16^4 + 324 = (16 * 22 + 18)( 16 * 10 + 18)$
$28^4 + 324 = (28 * 34 + 18)(28 * 22 + 18)$
$40^4 + 324 = (40 * 46 + 18)(40 * 34 + 18)$
so denominator $= (4^4+324)(16+324)(28+324)(40+324)$
$= ( 4 * (-2) + 18)(4 * 10 +18) (10 * 16 + 18) ( 16 * 22 + 18)(22 * 28 + 18) (28 * 34 + 18) (34 * 40 + 18) (40 *46 + 18)$
so ratio = $\frac{46* 52 +18}{4 * (-2) + 18} = 240$
or n = 240
check your answer
 
Albert said:
check your answer

mistake in last step
2410/10 or 241
 

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