Show that one and only one out of n, n + 2 or n +4 is divisible by 3, where n is any
positive integer.
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Show that one and only one out of n, n+2, n+4 is divisible by 3
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asked Oct 31, 2017 in Class XI Maths by aditya23 (-2,145 points)
Show that one and only one out of n, n+2, n+4 is divisible by 3
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answered Oct 31, 2017 by priya12 (-12,631 points)
Solution: We know that any positive integer is of the form 3q or 3q + 1 or 3q + 2 for some integer q & one and only one of these possibilities can occur
Case I : When n = 3q
In this case, we have,
n=3q, which is divisible by 3
n=3q
= adding 2 on both sides
n + 2 = 3q + 2
n + 2 leaves a remainder 2 when divided by 3
Therefore, n + 2 is not divisible by 3
n = 3q
n + 4 = 3q + 4 = 3(q + 1) + 1
n + 4 leaves a 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
Case II : When n = 3q + 1
In this case, we have
n = 3q +1
n leaves a reaminder 1 when divided by 3
n is not divisible by 3
n = 3q + 1
n + 2 = (3q + 1) + 2 = 3(q + 1)
n + 2 is divisible by 3
n = 3q + 1
n + 4 = 3q + 1 + 4 = 3q + 5 = 3(q + 1) + 2
n + 4 leaves a 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
Case III : When n = 3q + 2
In this case, we have
n = 3q + 2
n leaves remainder 2 when divided by 3
n is not divisible by 3
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 divsible by 3
n = 3q + 2
n + 4 = 3q + 2 + 4 = 3(q + 2)
n + 4 is divisible by 3
Thus, n + 4 is divisible by 3 but n and n + 2 are not divisible by 3 .
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answered Apr 8, 2018 by Manit Gera Basic (25 points)
Euclid's division Lemma any natural number can be written as: .
where r = 0, 1, 2,.........(a-1). and q is the quotient.
put a = 3: b = 3q+r and r = 0,1,2.
thus any number is in the form of 3q , 3q+1 or 3q+2.
case I: if n =3q
n is divisible by 3,
n+2 = 3q+2 is not divisible by 3.
n+4 = 3q+4 = 3(q+1)+1 is not divisible by 3.
case II: if n =3q+1
n = 3q+1 is not divisible by 3.
n+2 = 3q+1+2=3q+3 = 3(q+1) is divisible by 3.
n+4 = 3q+1+4 = 3q+5 = 3(q+1)+2 is not divisible by 3.
case III: if n = 3q+2
n =3q+2 is not divisible by 3.
n+2 = 3q+2+2 =3q+4 = 3(q+1)+1 is not divisible by 3.
n+4 = 3q+2+4 = 3q+6 = 3(q+2) is divisible by 3.
thus one and only one out of n , n+2, n+4 is divisible by 3.