Question

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

Answer 1

Answer:

The remainder is 3.

Step-by-step 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,

$(10^{k+1}-1) ( mod 4)$

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

$= 10(mod 4)\times (10^k-1)(mod 4 ) + 9 ( mod 4)$

$= (2\times 3)(mod 4) + 1$

$=2+1$

$=3$

Hence, our assumption is correct.

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