Divisibility Test Note 2

Find and prove the divisibility (so many i's) test for 11. Can you find a proof for $11^n$ ?

#NumberTheory #Divisibility #Sharky

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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}`	$\boxed{123}$

Comments

Easy to prove for 11,

$10 \equiv -1 \pmod{11}$

Thus , $10^{2n-1} \equiv -1 \pmod{11}$ ..... $\forall n\in \mathbb{N}$

and $10^{2n-2} \equiv 1 \pmod{11}$ ........ $\forall n\in \mathbb{N}$

We can write any number in decimal form as $a_n\times 10^n +a_{n-1}\times 10^{n-1} +... + a_2\times 10^2 + a_1\times 10+a_0$ where $a_i$ are actually digits from $0$ to $9$ .

Thus , if we take this whole thing modulo 11, it will become alternate $+$ and $-$ of the $a_i$ s and thus, we conclude that if "the difference between $k$ and $l$ is divisible by 11, the number is divisible by 11", where i define $k$ as sum of all digits having an even power of $10$ in the expanded form, and $l$ as the sum of all digits having an odd power of $10$ in the expanded form.

Aditya Raut - 6 years, 11 months ago

How about $11^n$ ?

Sharky Kesa - 6 years, 11 months ago

Math	Appears 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}`	$\boxed{123}$