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An eight-digit integer is formed by repeating a positive four-digit integer. For example, 25,632,563 or 60,786,078 are integers of this form. What is the greatest common divisor of all eight-digit integers of this form?

 Jun 21, 2022
 #1
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The answer is 11111111.

 Jun 21, 2022
 #2
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Note that we can write

 

a(10)^7  + b(10)^6 + c (10)^5  + d (10)^4  + a(10)^3 + b(10^2) + c(10) + d = 

 

a (10^7 + 10^3)  + b(10^6 + 10^2)  + c(10^5 + 10)  +  d(10^4 + 1)  =

 

a ( 10^3) (10^4 + 1)  + b (10^2) (10^4 +1)  + c(10)(10^4 + 1) + d (10^4 + 1)  =

 

(10^4 + 1)  [ a (10^3 )  + b (10^2) + c (10)  +  d  ]

 

The   gcd =  ( 10^4 + 1 )  = 10001

 

cool cool cool

 Jun 21, 2022

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