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Question

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

A
1
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B
7
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C
9
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D
13
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Solution

The correct option is A 1
777....(56 digits) = 7 * ( 111.....56 digits)
= 79 * ( 999....56 digits)
= 79 * ( 1000.... 56 0's - 1)
= 79 * ( 1056 - 1)
Now, we shall use Euler's theorem to find the remainder.

Divisor = 19.

So, ϕ(19) = 19 * (1 - 119)
= 18

Rem[5618] = 2
Therefore,
Rem[(79)(10561)19]=Rem[(79)102119]=Rem[79(1001)19]=Rem[(79)9919]=Rem[7719]
= 1

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