- #1
John112
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How do you prove that: Every perfect square is either a multiple of 3 or one more than a multiple of 3?
There is something very basic that I don't seem to understand in the question.
Here is proof that someone gave me:
Every natural integer p must fall in one of the following case:
♠ p is a multiple of 3 and can be expressed as p=3n
♣ p is 1 more than a multiple of 3 and can be expressed as p=3n+1
♥ p is 1 less than a multiple of 3 and can be expressed as p=3n–1
Then obviously:
♠ If p=3n:
p² = (3n)² = 9n²
and p is naturally a multiple of 3.
♣ If p=3n+1:
p² = (3n + 1)² = 9n² + 6n + 1 = 3(3n² + 2n) + 1
and p² is logically 1 more than a multiple of 3.
♥ If p=3n–1:
p² = (3n – 1)² = 9n² – 6n + 1 = 3(3n² – 2n) + 1
and p² is logically 1 more than a multiple of 3.
So we have proved that any perfect square is:
either a multiple of 3,
or one more than a multiple of 3.
But what I don't understand is: 0 and 1 are also perfect squares but they are not multiples of 3. Then how can we say that every perfect square is a multiple of 3 or one more than a multiple of 3? maybe it's when we set p = 3n is what's confusing me.
There is something very basic that I don't seem to understand in the question.
Here is proof that someone gave me:
Every natural integer p must fall in one of the following case:
♠ p is a multiple of 3 and can be expressed as p=3n
♣ p is 1 more than a multiple of 3 and can be expressed as p=3n+1
♥ p is 1 less than a multiple of 3 and can be expressed as p=3n–1
Then obviously:
♠ If p=3n:
p² = (3n)² = 9n²
and p is naturally a multiple of 3.
♣ If p=3n+1:
p² = (3n + 1)² = 9n² + 6n + 1 = 3(3n² + 2n) + 1
and p² is logically 1 more than a multiple of 3.
♥ If p=3n–1:
p² = (3n – 1)² = 9n² – 6n + 1 = 3(3n² – 2n) + 1
and p² is logically 1 more than a multiple of 3.
So we have proved that any perfect square is:
either a multiple of 3,
or one more than a multiple of 3.
But what I don't understand is: 0 and 1 are also perfect squares but they are not multiples of 3. Then how can we say that every perfect square is a multiple of 3 or one more than a multiple of 3? maybe it's when we set p = 3n is what's confusing me.
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