Question

Show that one and only one out of n, n + 4,n+8, n + 12, and n + 16

Is divisible by 5 Where n is any positive integer

Answer 1

Method 1:

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

Case I:

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 II:

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 result 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.

Method 2 :

Any positive integer is of the form 5q , 5q + 1 , 5q + 2,5q+3 and 5q+4

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 ]

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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 ]

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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 ]

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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]
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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