Srikanth has made a project on real numbers, where he finely expl… - Vidyadip

Question

Srikanth has made a project on real numbers, where he finely explained the applicability of exponential laws and divisibility conditions on real numbers. He also included some assessment questions at the end of his project as listed below. Answer them.
1. The statement 'One of every three consecutive positive integers is divisible by 3 ' is?
2. For what value of $n, 4^n$ ends in 0 ?
3. If $a$ is a positive rational number and $n$ is a positive integer greater than $I$, then for what value of $n, 4^n$ is a rational number?
Or
If n is any odd integer, then $n ^2-1$ is divisible by?

✓

Answer

1. Let three consecutive positive integers be $n , n +1$ and $n +2$.
We know that when a number is divided by 3, the remainder obtained is either 0 or 1 or 2 .
So, $n=3 p$ or $3 p+1$ or $3 p+2$, where pis some integer.
If $n=3 p$, then 2 is divisible by 3 .
If $n=3 p+1$, then $n+2=3 p+1+2=3 p+3=3(p+1)$ is divisible by 3 .
If $n=3 p+2$, then $n+1=3 p+2+1=3 p+3=3(p+1)$ is divisible by 3 .
So, we can say that one of the numbers among $n, n+1$ and $n+2$ is always divisible by 3.2 . For a number to end in zero it must be divisible by 5 , but $4^n=2^{2 n}$ is never divisible by 5 . So, $4^n$ never ends in zero for any value of $n$.
3. We know that product of two rational numbers is also a rational number.
So, $a^2=a \times a=$ rational number.
$a^3=a^2 \times a=$ rational number.
$a^4=a^3 \times a=$ rational number.
$a^n=a^{n-1} \times a=$ rational number.
or
Any odd number is of the form of $(2 k+1)$, where $k$ is any integer.
So, $n ^2-1=(2 k +1)^2-1=4 k ^2+4 k$
For $k=1,4 k^2+4 k=8$, which is divisible by 8 .
Similarly, for $k=2,4 k^2+4 k=24$, which is divisible by 8 .
And fork $=3,4 k^2+4 k=48$, which is also divisible by 8 .
So, $4 k^2+4 k$ is divisible by 8 for all integers $k$, i.e., $n^2-1$ is divisible by 8 for all odd values of $n$.

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