3(a12-1) - (b12-1)=13k ..............(by fermats theorem)
(a6-1) - (b6-1)=7k ..............(by fermats theorem)
hence,
a12-b12=91k
SHOW THAT a12- b12 IS DIVISIBLE BY 91 ,IF a AND b ARE BOTH PRIME TO 91
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2 Answers
1a f ( 7 ) ≡ 1 ( mod 7 )
( we can apply this because given that , a is coprime to 91 . Hence , a is co - prime to 7 and 13 also . )
where f ( P ) is Euler's totient function to determine number of co - prime numbers less than or equal to P .
f ( 7 ) = 6 , as 7 is a prime number .
So , a 6 ≡ 1 ( mod 7 ) ---> a 12 ≡ 1 ( mod 7 )
Similarly , b 12 ≡ 1 ( mod 7 )
Again , a f ( 13 ) ≡ 1 ( mod 13 )
f ( 13 ) = 12
Hence , a 12 ≡ 1 ( mod 13 )
Similarly , b 12 ≡ 1 ( mod 13 )
As , G . C . D . ( 7 , 13 ) = 1
So , a 12 ≡ 1 ( mod 7 x 13 ) ≡ 1 ( mod 91 ) also .
Similarly , b 12 ≡ 1 ( mod 91 )
Hence , a 12 - b 12 is always divisible by 91 .
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