71 Day Streak Special

$\large { \left( \sqrt { 10x+1 } +1 \right) }^{ 10x+1 }-{ \left( \sqrt { 10x+1 } -1 \right) }^{ 10x+1 }$

Find and generalise the last digit of the expression above to $x$ where $x$ is a whole number.

#Combinatorics

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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

What do you mean? Is $x$ supposed to be a fixed integer?

Or is the last digit of that expression independent of $x$ ? What if $x$ is a real number?

Calvin Lin Staff - 6 years ago

When \sqrt{10x-1} is rational,the last digit is always 2. Also ,x must be even. This would be because the of 10x+1 always ends with a 1, Some basic number theory says that sqrt(10x+1) ends with a 9 or 1. If 9,the expression is equivalent to (9+1)^(10x+1)-(9-1)^(10x+1) =0-8^(2x+1) =2 (mod 10) If 1, (1+1)^(10x+1)-(1-1)^(10x+1) =2^(2x+1)-0 =2-0 =2 mod(10). Hope this answers your question.

Timothy Wan - 6 years ago

Not true, The last digit of ${ \left( \sqrt { 91 } +1 \right) }^{ 91 }-{ \left( \sqrt { 91 } -1 \right) }^{ 91 }$ is 8

Joel Yip - 5 years, 12 months ago

I did state that the above is true if $\sqrt{10x+1}$ is rational. But now you mentioned it,I will try to generalize the cases. Thanks anyway.

Timothy Wan - 5 years, 12 months ago

20x

Hamza Iqbal - 5 years, 12 months ago

Joel Yip - 6 years 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}`	$\boxed{123}$