Answer:
The remainder is 3.
Step-by-step explanation:
We have to find out,
[tex]10^z-1(mod 4)=?\text{ where }z\geq 2[/tex]
If z = 2,
[tex]10^{2}-1=100-1=99[/tex]
∵ 99 ( mod 4 ) = 3,
Suppose,
[tex](10^{k}-1)(mod 4)=3\forall \text{ k is an integer greater than 2,}[/tex]
Now,
[tex](10^{k+1}-1) ( mod 4)[/tex]
[tex]= (10^k.10 - 10+9)(mod 4)[/tex]
[tex] = 10(mod 4)\times (10^k-1)(mod 4 ) + 9 ( mod 4)[/tex]
[tex]= (2\times 3)(mod 4) + 1[/tex]
[tex]=2+1[/tex]
[tex]=3[/tex]
Hence, our assumption is correct.
The remainder of [tex]10^z -1[/tex] divided by 4 is 3 where, z ≥ 2.
