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Home » Prove that one and only one out of n, n + 2 and n + 4 is divisible by 3, where n is any positive integer.
Prove that one and only one out of n, n + 2 and n + 4 is divisible by 3, where n is any positive integer.
CBSE | Class 10 | Maths | NCERT Exemplar | Real Numbers
Prove that one and only one out of n, n + 2 and n + 4 is divisible by 3, where n is any positive integer.

Solution:

According to Euclid’s division Lemma,

Let n be the positive integer

And b equals to 3

Where q is the quotient and r is the remainder n\text{ }=3q+r,

0<r<3 implies remainders may be 0, 1 and 2

‘n’ may be in the form of 3q, 3q+1, 3q+2

For n=3q

n+2=3q+2

n+4=3q+4

Only n is divisible by 3

For n\text{ }=\text{ }3q+1

n+2=3q=3

n+4=3q+5

Only n+2 is divisible by 3 in this case.

For n=3q+2

n+2=3q+4

n+4=3q+2+4=3q+6

Only n+4 is divisible by 3 in this case.

As a result, we can conclude that only one of n, n + 2, and n + 4 is divisible by three. Hence Proved

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