Question

Prove that every positive integer is of the form 3p,3p+1 and 3p+2 ,where p is any integer

Answer 1

To Prove :- every positive integer is of the form 3p,3p+1 and 3p+2 ,where p is any integer ?

Answer :-

Euclid's division Lemma states that for any two positive integers a and b there exist two unique whole numbers q and r such that :-

• a = bq + r, where 0 ≤ r < b .

Here,

• a = Dividend.
• b = Divisor.
• q = Quotient.
• r = Remainder.

so,

• The values r can take = 0 ≤ r < b .

we have given that, p is any integer in the form 3p, 3p + 1 and 3p + 2 .

so,

→ b = 3 .

then,

→ Possible values of r = 0, 1, and 2 . { since 0 ≤ r < 2 . }

therefore, when r = 0,

→ a = bq + r

→ a = 3p + 0

→ a = 3p

when r = 1,

→ a = bq + r

→ a = 3p + 1

when r = 2,

→ a = bq + r

→ a = 3p + 2 .

therefore, we can conclude that, any positive integer a can be written in the form of 3p, 3p+1 and 3p+2 ,where p is any integer .

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