Question

For every positive integer a , find a composite number n such that n|a^n-a.?
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Answer 1

Answer:

If the density of an object is more than the density of a liquid, then it sinks in the liquid. On the other hand, if the density of an object is less than the density of a liquid, then it floats on the surface of the liquid.

Density =

Volume

Mass

=

20

50

=2.5 g/cm

3

Hence, The object will sink.

Answer 2

Given,

A positive integer is termed as a.

Solution,

Assume n to be a natural number.

So, the number is given as

$\[\begin{array}{l} \Rightarrow 2 + \left( {n + 1} \right)!,3 + \left( {n + 1} \right)!,4 + \left( {n + 1} \right)!\\ \Rightarrow n + \left( {n + 1} \right)!,\left( {n + 1} \right) + \left( {n + 2} \right)! - - - - \left( 1 \right)\end{array}\]$

So from the above equation (1),

$\[ \Rightarrow \left( 1 \right) \in n|{a^n} - a\]$

Divide the number by $\[2\].$

$\[2 + \left( {n + 1} \right)!\]$ is a composite number.

$\[ \Rightarrow n! = 1 \times 2 \times 3 \times 4... \times n\]$

Therefore, it gcd$\[\left( {n \ne m} \right) > 1\]$, then $\[n! + m\]$is composite with each m in $\[1 < m < n\]$ ensuring them of composite form$\[n! + 2\]$ to $\[n! + n\]$such that $\[n|{a^n} - a\].$