Question

1+5555=85+2222 verify the answer is in proportion​

Answer 1

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.