Theorem: a
n
−b
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)
=(4
2
)
1111
+(3
5
=(16)
+(243)
.
Now (16)
is divisible by 16+243 or it is divisible by 259, which is a multiple of 7.
Hence the remainder when (5555)
+(2222)
is divided by 7 is zero.
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Verified answer
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.