Question

Prove that 2⁵⁵⁵⁵ + 5²²²² is divisible by 7?​

Answer 1

Answer:

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Step-by-step explanation:

Theorem: a  

n

−b  

n

 is divisible by a−b

The remainders when 5555 and 2222 are divided by 7 are 4 and 3 respectively.

Hence, the problem reduces to finding the remainder when (4)  

2222

+(3)  

5555

 is divided by 7.

Now (4)  

2222

+(3)  

5555

=(4  

2

)  

1111

+(3  

5

)  

1111

=(16)  

1111

+(243)  

1111

 .

Now (16)  

1111

+(243)  

1111

 is divisible by 16+243 or it is divisible by 259, which is a multiple of 7.

Hence the remainder when (5555)  

2222

+(2222)  

5555

 is divided by 7 is zero.