Determine-the-integer-which-can-be-written-N-xyz-7-zyx-11-

Question Number 119440 by mathocean1 last updated on 24/Oct/20

Determine the integer which can

be written : N = \overset{―}{xyz}^{7} = \overset{―}{zyx}^{11}

Commented by mathocean1 last updated on 24/Oct/20

Please can you detail sir

Answered by mr W last updated on 25/Oct/20

x, y, z a r e d i g i t s o f a n i n t e g e r o f b a s e 7,

\Rightarrow 0 ⩽ x, y, z ⩽ 6

b e s i d e s, x ⩾ 1, z ⩾ 1 .

{(x y z)}_{7} = 49 x + 7 y + z

{(z y x)}_{11} = 121 z + 11 y + x

\Rightarrow 49 x + 7 y + z = 121 z + 11 y + x

\Rightarrow 48 x - 4 y = 120 z

\Rightarrow 12 x - y = 30 z

y m u s t b e a m u l t i p l e o f 6, s a y y = 6 Y

\Rightarrow 2 x - Y = 5 z

Y = 0 o r 1

w i t h Y = 0 :

2 x = 5 z

\Rightarrow x = 5, z = 2

w i t h Y = 1 :

2 x - 1 = 5 z

\Rightarrow x = 3, z = 1

t h e s o l u t i o n s a r e :

x = 5, Y = 0 (i . e . y = 0), z = 2

\Rightarrow N = {(502)}_{7} = {(205)}_{11} = 247

x = 3, Y = 1 (i . e . y = 6), z = 1

\Rightarrow N = {(361)}_{7} = {(163)}_{11} = 190

Commented by mathocean1 last updated on 25/Oct/20

Thank you sir .