Question

The number (10ⁿ-1) is divisible by 11 for ……….
plz help me ​

Answer 1

Using Binomial expansion (10)=(11–1)
(
10
n
)
=
(
11
–
1
)
n

Note: In the above binomial expansion all terms except the last one are all divisible by 11:

The term not divisible by 11
11
is (−1)
(
−
1
)
n

∴10−1|11=(−1)−1|11
∴
10
n
−
1
|
11
=
(
−
1
)
n
−
1
|
11

For divisibility by 11 : 10−1|11=0⟹(−1)−1|11=0
10
n
−
1
|
11
=
0
⟹
(
−
1
)
n
−
1
|
11
=
0


The above equation will hold good only if
n
is even