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Number Theory Level 1

$\large {1}^{1}×{2}^{2}×{3}^{3}×{4}^{4}×{5}^{5}×\dots×{99}^{99}×{100}^{100}$

What will be the unit's digit in the above expression?

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The answer is 0.

2 solutions

Ashish Menon
Jun 13, 2016

${100}^{100}$ is a multiple of $10$ so the whole product is a multiple of $10$ thus it ends with $\color{#3D99F6}{\boxed{0}}$ .

Brilliant!

Abhay Tiwari - 5 years ago

Thanks! :)

Ashish Menon - 5 years ago

Exactly that's what I did bro! (+1)!

Rishabh Tiwari - 5 years ago

The toughest part would be asking us to find the number of Trailing Zeros.

Samara Simha Reddy - 5 years ago

Nice thought :)

Abhay Tiwari - 5 years ago

Dude. Don't try it. It takes a lot of time.

Samara Simha Reddy - 5 years ago

@Samara Simha Reddy – Ha ha, I know ;). It will take me Maybe 2 days. Still nice thought (+1)

Abhay Tiwari - 5 years ago

@Abhay Tiwari – Thanks. Even I ll work on it.

Samara Simha Reddy - 5 years ago

No, The toughest part would be asking us to find the first non-zero digit from right. As for the trailing zeroes question try this

Ravneet Singh - 5 years ago

It takes time for me Let me work on it.

Samara Simha Reddy - 5 years ago

Rushikesh Jogdand
Jul 1, 2016

Relevant wiki: Modular Arithmetic - Multiplication

We are required to find ${1}^{1}×{2}^{2}×{3}^{3}×{4}^{4}×{5}^{5}×\dots×{99}^{99}×{100}^{100}\mod{10}$ $100\equiv 0 \mod{10}$ $\therefore 100^{100}\equiv 0^{100} \equiv 0\mod{10}$ $\therefore {1}^{1}×{2}^{2}×{3}^{3}×{4}^{4}×{5}^{5}×\dots×{99}^{99}×{100}^{100}\equiv {1}^{1}×{2}^{2}×{3}^{3}×{4}^{4}×{5}^{5}×\dots×{99}^{99}×0\equiv\boxed{0} \mod{10}$

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