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Divisibility By 13

I just wanted to know that If "111111111....... 54 times" is divisible by 13 or not??

Can anyone tell me??

And also how or how not??????

Note by Anand Raj
2 years, 11 months ago

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\( \underbrace{111\ldots111}_{\text{54 1s}} \)

\( = 111111(10^{49} + 10^{43} + \ldots + 10^7 + 1) \)

\( = 1001 \times 111(10^{49} + 10^{43} + \ldots + 10^7 + 1) \)

\( = 13 \times 77 \times 111(10^{49} + 10^{43} + \ldots + 10^7 + 1) \)

So, \( 13|\underbrace{111\ldots111}_{\text{54 1s}} \)

More generally, \( 13|\underbrace{111\ldots111}_{\text{n 1s}} \) if \( 6|n \) Siddhartha Srivastava · 2 years, 11 months ago

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@Siddhartha Srivastava Thanx Dude.............. Anand Raj · 2 years, 11 months ago

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hmmmmmmmmmm I know that method but I Wanted to know is there any trick for 111111..... Anand Raj · 2 years, 11 months ago

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Add 4 times the last digit to the remaining truncated number. Repeat the step as necessary. If the result is divisible by 13, the original number is also divisible by 13.

Check for 3146:: 314+ (46) = 338:: 33+(48) = 65. Since 65 is divisible by 13, the original no. 3146 is also divisible.

I guess that would take years to check whether 11111.. is divisible but according to my knowledge, it is not divisible by 13. Aryan Gaikwad · 2 years, 11 months ago

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