For any given even number $\sqrt{A}$ $sqrtA$ that has $m$ $m$ digits and $A$ $A$ with $n$ $n$ digits, is there a relationship between $m$ $m$ and $n$ $n$ ?

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For any given even number $\sqrt{A}$ $sqrtA$ that has $m$ $m$ digits and $A$ $A$ with $n$ $n$ digits, is there a relationship between $m$ $m$ and $n$ $n$ ?

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Answer 1 · 2017-03-23T12:47:33

if $n$ $n$ is even, $m = \frac{n}{2}$ $m=n/2$
if $n$ $n$ is odd, $m = \frac{n + 1}{2}$ $m=(n+1)/2$

where $m$ $m$ is the number of digits in $\sqrt{A}$ $sqrtA$

Explanation:

Let's first look at a few terms and see if we can spot a pattern (I'm including the variable $m$ $m$ which is the number of digits in $\sqrt{A}$ $sqrtA$ ):

$((A, n,sqrtA, m),(4,1,2,1),(16,2,4,1),(36,2,6,1),(64,2,8,1),(100,3,10,2),(144,3,12,2),(vdots, vdots, vdots, vdots))$

Since $A$ is even, then $sqrtA$ will also be even. And so what I'll do next is look at what values of $A$ we generate for different values of $sqrtA$ with different $m$ :

$((A, n,sqrtA, m),(4,1,2,1),(16,2,4,1),(vdots, vdots, vdots, vdots),(64,2,8,1),(100,3,10,2),(vdots, vdots, vdots, vdots),(900,3,30,2),(1024,4,32,2),(vdots, vdots, vdots, vdots),(9604,4,98,2),(10000,5,100,3))$

And so we can see relationships:

$n=1, m=1$
$n=2, m=1$
$n=3, m=2$
$n=4, m=2$
$n=5, m=3$

and so on.

And so we can calculate $m$ for any $n$ , where,

if $n$ is even, $m=n/2$
if $n$ is odd, $m=(n+1)/2$

Let's test it out:. Let's say $sqrtA=45678, m=5=>A=2086479684, n=10 color(green)sqrt$

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For any given even number $\sqrt{A}$ $sqrtA$ that has $m$ $m$ digits and $A$ $A$ with $n$ $n$ digits, is there a relationship between $m$ $m$ and $n$ $n$ ?

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Explanation:

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For any given even number $\sqrt{A}$ $sqrtA$ that has $m$ $m$ digits and $A$ $A$ with $n$ $n$ digits, is there a relationship between $m$ $m$ and $n$ $n$ ?