Question

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ANSWER THIS ATTACHED QUES...​

Answer 1

$\mathcal{\huge{\boxed{ANSWER\::-}}}$

$\large{\underline{\underline{Given\::-}}}$

>> n is a positive integer

>> Numbers :- n , n+4 , n+8 , n+12, n+16

• To show that only one of the number above is divisible by 5



$\underline{\underline{Conditions\: required\::-}}$

>> Any number is divisible by 5 if it is in the form of 5(x) , where x is any integer

(for the integer value)

>>Then the numbers should be multiple of 5



$\underline{\underline{Steps\:to\: Show\::-}}$

•Let us consider that out of Numbers :-

n , n+4 , n+8 , n+12, n+16

n is divisible by 5 = 5x



•Then we will check divisibility of other numbers

for

> n+4 = 5x + 4 [which is not a multiple of 5]

> n+8 = 5(x+1) + 3 [which is not a multiple of 5]

> n+12 = 5(x+2) + 2 [which is not a multiple of 5]

> n+16 = 5(x+3) + 1 [which is not a multiple of 5]




>> Similarly we can also conclude that if one them is a multiple of "5" then others will not fulfill the condition to be divisible by 5



>> Hence ,Shown that only one Number out of Numbers :- n , n+4 , n+8 , n+12, n+16 is divisible by 5 .

$\sf{\green{\underline{\underline{\large{Hope\: It \: Helps}}}}}$

Answer 2

SOLUTION

Any positive integer will be of form 5q,5q+1,5q+2,5q+3,or 5q+4

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

When, 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) will 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 can check the results for 5q+2, 5q+3 and 5q+4.

In each case only one out of n, n+2,n+4, n+8, n+16 will be divisible by 5.

HOPE it helps ✔️