×

# 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
4 years ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

$$\underbrace{111\ldots111}_{\text{54 1s}}$$

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

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

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

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

More generally, $$13|\underbrace{111\ldots111}_{\text{n 1s}}$$ if $$6|n$$

Thanx Dude..............

- 4 years ago

hmmmmmmmmmm I know that method but I Wanted to know is there any trick for 111111.....

- 4 years ago

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.

- 4 years ago