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Algebra Level 2

If y = a 1 b 2 c 3 d 4 . . . . . t 20 v 21 w 22 x 23 y = a^1 b^2 c^3 d^4.....t^{20}v^{21}w^{22} x^{23} and z = a 23 b 22 c 21 d 20 . . . t 4 v 3 w 2 x 1 z = a^{23} b^{22} c^{21} d^{20} ...t^{4} v^{3} w^{2} x^{1} , find the value of a b c d . . . t v w x abcd...tvwx suppose y z = 1 7 72 yz=17^{72} .


The answer is 4913.

Noel Lo by Noel Lo , 24, Singapore

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1 solution

Noel Lo
May 7, 2015

y z = a 1 b 2 c 3 d 4 . . . . . t 20 v 21 w 22 x 23 × a 23 b 22 c 21 d 20 . . . t 4 v 3 w 2 x 1 yz = a^1 b^2 c^3 d^4.....t^{20}v^{21}w^{22} x^{23} \times a^{23} b^{22} c^{21} d^{20} ...t^{4} v^{3} w^{2} x^{1}

= ( a 1 a 23 ) ( b 2 b 22 ) ( c 3 c 21 ) ( d 4 d 20 ) . . . ( t 20 t 4 ) ( v 21 v 3 ) ( w 22 w 2 ) ( x 23 x 1 ) = (a^1 a^{23}) (b^2 b^{22}) (c^3 c^{21})(d^4 d^{20})... (t^{20} t^4) (v^{21} v^3) (w^{22} w^2) (x^{23} x^1)

= ( a b c d . . . t v w x ) 24 = (abcd...tvwx)^{24}

( a b c d . . . t v w x ) 24 = 1 7 72 (abcd...tvwx)^{24} = 17^{72}

( a b c d . . . t v w x ) 24 = ( 1 7 3 ) 24 (abcd...tvwx)^{24} = (17^3)^{24}

a b c d . . . t v w x = 1 7 3 = 4913 abcd...tvwx = 17^3 = \boxed{4913}

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