Launs

Algebra Level 3

ln ( x ) = ln ( 1 ) + ln ( 2 ) + ln ( 3 ) + . . . + ln ( y ) x y = 720 x + y = ? \begin{aligned} \ln { \left( x \right) } & = & \ln { \left( 1 \right) + } \ln { \left( 2 \right) + } \ln { \left( 3 \right) + } ...+\ln { \left( y \right) } \\ \frac x y & = & 720 \\ x + y & = & \ ? \\ \end{aligned}


The answer is 5047.

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2 solutions

Aniket Verma
Apr 2, 2015

ln x = ln 1 + ln 2 + ln 3 + ln y \ln { x } =\quad \ln { 1 } +\ln { 2 } +\ln { 3 } \dots \dots +\ln { y }

ln x ln y = ln { 1 × 2 × 3 × 4....... × ( y 1 ) } \implies \ln { x\quad -\quad \ln { y } } =\quad \ln { \{ 1\times 2\times 3\times 4.......\times (y-1) } \}

ln x y = ln { ( y 1 ) ! } \implies \ln { \dfrac { x }{ y } } =\quad \ln { \{ (y-1)!\} }

x y = ( y 1 ) ! = 720 = 6 ! \implies \dfrac{x}{y} = (y-1)! = 720 = 6!

y 1 = 6 , y = 7 y-1 = 6,~ \therefore y=7

& x y = 720 , x = 720 × y = 5040 \& \because \dfrac{x}{y} = 720~, ~\therefore ~x = 720\times y = 5040

x + y = 5040 + 7 = 5047 \therefore x+y = 5040 + 7= 5047

John Aries Sarza
Apr 2, 2015

x/y =6! Thus y=7 (next of 6) x=7*720=5040 x+y= 5047

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