[Math] Find the Remainder when $792379237923$…upto 400 digits is divided by $101$

Question

Find the Remainder when $792379237923\ldots$upto 400 digits is divided by $101$?

MyApproaach

when ($792379237923\ldots$400 digts)/$101$=

I learned this approach that I have to calculate(let say U)=Is the sum all of all the alternate groups starting with the rightmost

and (let say)Th=Is the the sum all of all the alternate groups starting with the second rightmost

Rem(U-Th)/$101$=?

But I am not following how to calculate U and Th

Can anyone guide me how to approach this problem?

Answer 1

Let $N$ be the number. Then we really want to find $N \pmod {101}$.

Note that $N=7923\cdot10^{396}+7923\cdot10^{392}+\cdots+7923$.

Next note that $7923\equiv 45 \pmod {101}$

Also, for example, $10^{100}=100^{50}\equiv(-1)^{50}\equiv 1 \pmod {101}$

We get the same result for each term, and there are $100$ of these terms so $N\equiv 45\cdot 100\equiv 56$.