Theorem:
A number is a square if and only if it has an odd number of factors.
Proof:
If the number is not a square, then the factors all come in pairs,
thus there are an even number of them.
If the number is a square, all of the factors are in pairs
except the square root.
So the total number of factors is odd. QED.
Let us see few eamples.
Take 35 and 36. 36 is a perfect square and 35 is not.
Factors of 35:
1,5,7, 35 (4 factors, even)
Factors of 36:
1,2,3,4,6,9,12, 18,36 (totally 9 which is ODD)
Pairs of factors:
1x36, 2x18, 3x12, 4x9, 9x4, 12x3, 18x2, 36x1 (totally 8 pairs) and
6 alone (the square root of 36) is left unpaird.
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A number is a square if and only if it has an odd number of factors.
Proof:
If the number is not a square, then the factors all come in pairs,
thus there are an even number of them.
If the number is a square, all of the factors are in pairs
except the square root.
So the total number of factors is odd. QED.
Let us see few eamples.
Take 35 and 36. 36 is a perfect square and 35 is not.
Factors of 35:
1,5,7, 35 (4 factors, even)
Factors of 36:
1,2,3,4,6,9,12, 18,36 (totally 9 which is ODD)
Pairs of factors:
1x36, 2x18, 3x12, 4x9, 9x4, 12x3, 18x2, 36x1 (totally 8 pairs) and
6 alone (the square root of 36) is left unpaird.
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