Question

Show that only one out of n, n + 4, n + 8, n + 12, n + 16 is divisible by 5, when n is a positive integer.

Answer 1

Answer (it is very long and this is the shortest it can get):

by Euclid's division lemma, a=bq+r, 0≤r<b

Let,

a= n

b= 5

r= 0,1,2,3,4

Case 1 (r=0):-

n=5q     {divisible by 5}

n+4=5q+4   {not divisible by 5}

n+8=5q+8    {not divisible by 5}

n+12=5q+12   {not divisible by 5}

n+16=5q+16   {not divisible by 5}

only n is divisible by 5 when r is equal to 0.

Case 2 (r=1):-

n=5q+1   {not divisible by 5}

n+4=5q+1+4

     =5q+5   {divisible by 5}

n+8=5q+1+8

      =5q+9   {not divisible by 5}

n+12=5q+1+12

       =5q+13   {not divisible by 5}

n+16=5q+1+16

       =5q+17   {not divisible by 5}

only n+4 is divisible by 5 when r is equal to 1

Case 3 (r=2):-

n=5q+2   {not divisible by 5}

n+4=5q+2+4

     =5q+6   {not divisible by 5}

n+8=5q+2+8

     =5q+10   {divisible by 5}

n+12=5q+2+12

       =5q+14   {not divisible by 5}

n+16=5q+2+16

       =5q+18   {not divisible by 5}

only n+8 is divisible by 5 when r is equal to 2

Case 4 (r=3):-

n=5q+3   {not divisible by 5}

n+4=5q+3+3

     =5q+6   {not divisible by 5}

n+8=5q+3+8

     =5q+11   {not divisible by 5}

n+12=5q+3+12

      =5q+15   {divisible by 5}

n+16=5q+3+16

       5q+19   {not divisible by 5}

only n+12 is divisible by 5 when r is equal to 3

Case 5 (r=4):-

n=5q+4   {not divisible by 5}

n+4=5q+4+4

     =5q+8   {not divisible by 5}

n+8=5q+4+8

     =5q+12   {not divisible by 5}

n+12=5q+4+12

       =5q+16   {not divisible by 5}

n+16=5q+4+16

       =5q+20   {divisible by 5}

only n+16 is divisible by 5 when r is equal to 4

hope this helps

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