Question

Let z≥2. What is the remainder of 10^z −1 divided by 4?

Answer 1

The remainder is 3.

Explanation:

We have to find out,

`10^z-1(mod 4)=?\text{ where }z\geq 2`

If z = 2,

`10^(2)-1=100-1=99`

∵ 99 ( mod 4 ) = 3,

Suppose,

`(10^(k)-1)(mod 4)=3\forall \text{ k is an integer greater than 2,}`

Now,

`(10^(k+1)-1) ( mod 4)`

`= (10^k.10 - 10+9)(mod 4)`

`= 10(mod 4)* (10^k-1)(mod 4 ) + 9 ( mod 4)`

`= (2* 3)(mod 4) + 1`

`=2+1`

`=3`

Hence, our assumption is correct.

The remainder of
`10^z -1` divided by 4 is 3 where, z ≥ 2.