Question

What is the remainder when 77,777... up to 56 digits is divided by 19?

• A. 1
• B. 7
• C. 9
• D. 13

Answer 1

The correct option is A 1

777....(56 digits) = 7 * ( 111.....56 digits)
= $\frac{7}{9}$ * ( 999....56 digits)
= $\frac{7}{9}$ * ( 1000.... 56 0's - 1)
= $\frac{7}{9}$ * ( ${10}^{5}6$ - 1)
Now, we shall use Euler's theorem to find the remainder.

Divisor = 19.

So, $\varphi$(19) = 19 * (1 - $\frac{1}{19}$)
= 18

Rem$\left[\frac{56}{18}\right]$ = 2
Therefore,
$Rem\left[{\left(\frac{7}{9}\right)}^{\ast }\frac{({10}^{56}-1)}{19}\right]=Rem\left[{\left(\frac{7}{9}\right)}^{\ast }\frac{{10}^2-1}{19}\right]=Rem\left[{\frac{7}{9}}^{\ast }\frac{(100-1)}{19}\right]=Rem\left[{\left(\frac{7}{9}\right)}^{\ast }\frac{99}{19}\right]=Rem\left[\frac{77}{19}\right]$
= 1