Question

prove that one and only one out of n, n+2, n+4is divisible by 3. best ans will get brainliest?????

Answer 1

The answer is as follows:-


When n = 3q, we have

n= 3q, which is divisible by 3

=>n+2 = 3q+2

n+2 leaves remainder 2 when divided by 3

Again, n = 3q

n+4 = 3q+4=3(q+1)+1

n+4 leaves remainder 1 when divided by 3

n+4 is not divisible by 3.

Thus, n is divisible by 3 but n+2 and n+4 are not divisible by 3.

 

When n = 3q+1, we have

n= 3q+1

When divided by 3, n leaves remainder 1.

=> n is divisible by 3

Now, n = 3q+1

n+2 = (3q+1)+2=3(q+1)

n+2 is divisible by 3.

Again, n = 3q+1

n+4 = 3q+1+4=3q+5=3(q+1)+2

n+4 leaves remainder 2 when divided by 3

n+4 is not divisible by 3.

Thus, n+2 is divisible by 3 but n and n+4 are not divisible by 3.

 

Similarly,When we have n + 3q+2,

n= 3q+2

n leaves remainder 2 when divided by 3.

n is not divisible by 3.

Now, n = 3q+2

n+2 = 3q+2+2=3(q+1)+1

n+2 leaves remainder 1 when divided by 3

n+2 is not divisible by 3.

Now, n = 3q+2

n+4 = 3q+2+4=3(q+2)

n+4 is divisible by 3.

Therefore, n+4 is divisible by 3 but n and n+2 are not divisible by 3.

Hence it is justified that one and only one out of n, n+2 , n+4 is divisible by 3 in each case.

Answer 2

Hey there !!

Any positive integer is of the form 5q , 5q + 1 , 5q + 2

here ,
b = 5
r = 0 , 1 , 2 , 3 , 4

when r = 0 , n = 5q
n = 5q ----> divisible by 5 ===> [1]
n + 4 = 5q + 4 [ not divisible by 5 ]
n + 8 = 5q + 8 [ not divisible by 5 ]
n + 6 = 5q + 6 [ not divisible by 5 ]
n + 12 = 5q + 12 [ not divisible by 5 ]

-------------------------------------------

when r = 1 , n = 5q + 1
n = 5q + 1 [ not divisible by 5 ]
n + 4 = 5q + 5 = 5 [q+ 1] ----> divisible by 5 ===> [2]
n + 8 = 5q + 9 [ not divisible by 5 ]
n + 6 = 5q + 7 [ not divisible by 5 ]
n + 12 = 5q + 13 [ not divisible by 5 ]

----------------------------------------------
when r = 2 , n = 5q + 2
n = 5q + 2 [ not divisible by 5 ]
n + 4 = 5q + 6 [ not divisible by 5 ]
n + 8 = 5q +10 = 5 [q + 2 ] ---> divisible by 5 ====> [3]
n + 6 = 5q +8 [ not divisible by 5 ]
n + 12 = 5q + 14 [ not divisible by 5 ]

----------------------------------------
when r = 3 , n = 5q + 3
n = 5q + 3 [ not divisible by 5 ]
n + 4 = 5q + 7 [ not divisible by 5 ]
n + 8 = 5q + 11 [ not divisible by 5 ]
n + 6 = 5q + 9 [ not divisible by 5 ]
n + 12 = 5q + 15 = 5 [ q + 3 ] ---> divisible by 5 ====> [4]
----------------------------------------------------
when r = 4 , n = 5q + 4
n = 5q + 4 [ not divisible by 5 ]
n + 4 = 5q + 8 [ not divisible by 5 ]
n + 8 = 5q + 12 [ not divisible by 5 ]
n + 6 = 5q + 10 = 5 [ q + 2 ] ---> divisible by 5 ====> [5]
n + 12 = 5q + 16 [ not divisible by 5 ]

from (1) , (2), (3) , (4) , (5) its clear that one and only one out of n, n+4, n+8, n+12 and n+6 is divisible by 5