Question

Show that one and only one out of p,p+2,p+4 is divisible by 3

Answer 1

Hey Friend !

Let "p" be any positive integer , that can be written in the form 3q , 3q+ 1 and 3q+ 2


Case I :- [ when p = 3q ]

p = 3q -----> divisible by 3 ====> (1)

p + 2 = 3q + 2 ---> not divisible by 3

p + 4 = 3q + 4 -----> not divisible by 3
-------------------------------------------------------------


Case II :- [ when p = 3q + 1 ]

p = 3q + 1 ----> not divisible by 3

p + 2 = 3q + 1 + 2 = 3q + 3 -----> divisible by 3 ====> (2)

p + 4 = 3q + 1 + 4 = 3q + 5 ----> not divisible by 3


Case II :- [ when p = 3q + 2 ]

p = 3q + 2 ----> not divisible by 3

p + 2 = 3q + 2 + 2 = 3q + 4 ----> not divisible by 3

p + 4 = 3q + 2 + 4 = 3q + 6 -----> divisible by 3 ====> (3)


From (1) , (2) and (3) ,
we get to know that one and only one out of p,p+2,p+4 is divisible by 3

Answer 2

dude we need to find one and only one out of p , p+2 , p+4 is divisible by 3 so i

will make some condition to solve ur question

if p is divisible by 3

we can let p = 3T  for some integers !

here we can put value of p in p+2

we get 3t +2 which means it is not divisible by 3

again we have p+4

which means 3t + 4 which means it is not divisible by 3 for better

understanding i will reduce this  = == = = = = >  3t+4= 3t+3 + 1 = 3(t+1) +1

which means it is not divisible by 3
 
let think if p +2  = 3t

now putting value in p +4

we get  p +2 + 2 which means (3t +2 ) again it is not divisible by 3

let think p +4  is divisible by 3  so p +4 => 3t

now we get here p= 3t -4

for better understanding  i will reduce this equation = p = 3t-3 -1 which

3(t-1)-1  is not perfectly divisible by 3 which means only p is divisible by 3

hence prove

HOPES THIS HELPS YOU !

cheers ^_^