Question

One and only one out of $n,n+4,n+8,n+12andn+16$ is ......(where n is any positive integer)

• A. Divisible by 5
• B. Divisible by 4
• C. Divisible by 10
• D. Divisible by 12

Answer 1

The correct option is A Divisible by 5

We know that any positive integer is of the form $5q,5q+1or5q+2,5q+3or5q+4$ for some integer $q$.
Case-1
If $n=5q$
n is divisible by $5$
Now,
$n=5q$
$=>n+4=5q+4$
The number $(n+4)$ will leave remainder $4$ when divided by $5$.
Again,
$n=5q$
$=>n+8=5q+8$
$=5(q+1)+3$
The number $n+8$ will leave remainder $3$ when divided by $5$
Again,
$n=5q$
$=>n+12=5q+12$
$=5(q+2)+2$
The number $n+12$ will leave remainder $2$ when divided by $5$
Again,
$n=5q$
$=>n+16=5q+16$
$=5(q+3)+1$
The number $n+16$ will leave remainder $1$ when divided by $5$
Case=2
$n=5q+1$
The number $n$ will leave remainder $1$ when divided by $5$
Now,
$n=5q+1$
$=>n+2=5q+3$
The number $n+2$ will leave remainder $3$ when divided by $5$
Again,
$n=5q+1$
$=>n+4=5q+5$
$=5(q+1)$
The number $n+4$ wil be divisible by $5$
Again,
$n=5q+1$
$=>n+8=5q+9$
$=5(q+1)+4$
The number $n+8$ will leave remainder $4$ when divided by $5$
Again,
$n=5q+1$
$=>n+12=5q+13$
$=5(q+2)+3$
The number $n+12$ will leave remainder $3$ when divided by $5$
Again,
$n=5q+1$
$=>n+16=5q+17$
$=5(q+3)+2$
The number $n+16$ will leave remainder $2$ when divided by $5$
Similarly we check for $5q+2,5q+3,5q+4$
In each case only one out of $n,n+2,n+4,n+8,n+16$ will be divisible by $5$