Remainder???

Hey Brilliantians!!

I just wanted to know how do you solve problems related to finding remainders? Like example- Find the remainder when 1000! is divided by 1000. Please can anyone tell me how to do these kind of problems

#Algebra #NumberTheory

Easy Math Editor

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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}`	$\boxed{123}$

Comments

could anyone please tell me why my advice got downvoted? It'd be helpful to know this.

mathh mathh - 6 years, 10 months ago

My also comment got downvoted.

Anuj Shikarkhane - 6 years, 10 months ago

See this post. Read about Euler's Theorem and its special case called Fermat's Little Theorem. Also read about Wilson's theorem, Chinese Remainder Theorem (Extended Euclidean Algorithm is related to it, read about it too.). They are all very useful and worth learning if you want to master modular arithmetic.

The less useful one is the Euler's Theorem's equivalent but with the Carmichael's function $\lambda$ used instead of the Euler's Totient function $\phi$ . We have $\lambda\le \phi$ , which means the Carmichael's function is 'stronger' sometimes - we could occasionally reduce the exponent more than using the Euler's totient function. You can read more about it and a lot of other stuff here. It is not a popular theorem, though.

mathh mathh - 6 years, 10 months ago

Thanks I would do it.It's a little tough but I can manage

Anuj Shikarkhane - 6 years, 10 months ago

@Calvin Lin I can't see why my comment got downvoted. If you have an idea why, it'd be great to know.

mathh mathh - 6 years, 10 months ago

Hey there...1000! is divisible by 1000...so remainder is zero...but for more complex problems, u should look up the results and theoroms related to modular mathematics..it should help..

Chandan Mahapatra - 6 years, 10 months ago

How can you say that 1000! is divisible by 1000??

Anuj Shikarkhane - 6 years, 10 months ago

$1000!=1000\times 999\times 998\times \cdots\times 2\times 1$ by the definition of a factorial.

mathh mathh - 6 years, 10 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}`	$\boxed{123}$