Is it always, sometimes or never true that a square number has an even number of factors? Explain why?
Always

Sometimes

Never
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answers:
Krishnamurthy say: Yes, it is a fact that a number is a square if and only if it has an odd number of divisors. You'd prove this using two things. First, you need the formula for the number of divisors of an integer. This formula is in terms of the exponents in the integer's prime factorization. The second thing that's needed is the fact that an integer is a square if and only if every exponent in its prime factorization is even.
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Jeffrey K say: Never. This is easy to explain. All factors come in pairs that multiply to give your number except for one factor. The square root of your number pairs up with itself but you only count it once. So a square number always has an odd number of factors.
36: 1,36; 2,18; 3,12; 4,9; 6
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nbsale say: If the prime factorization of n = p^a X q^b X ...
then the number of factors is (a+1)(b+1)....

That's because when you create one of the factors, you have a+1 choices for powers of p:
1 p p^2 ... p^a
and b+1 choices for powers of q, etc.

Since a square number has even powers of its prime factors, the number of factors will be something like (2n +1)(2m+1)....
That's a product of odd numbers, so it will have an odd number of factors.

Look at a few examples:
4 has 3 factors: 1 2 4

36 has 9 factors
Choose one of 1 2 4
and one of 1 3 9
to get these nine factors: 1 3 9 2 6 18 4 12 36.
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Steve A say: If you mean different prime factors, then no. The only prime factor of k^2 is k. So always odd.
If you mean factors (numbers > 1 that divide into k^2 with no remainder), the number is always even. Factors of 9 are 3,9. Factors of 16 are 2,4,8,16.
If you include 1, always odd.
If you mean factorization, k^2 factor to k*k. If k can be factored further (say by two), both k's have that characteristic.
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TomV say: Always.
Let the number be n = k², then n has the repeated factor k. If k has j factors, then n has 2j factors which will always be an even number whether j is even or odd.
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alex say: Do you mean Prime factors ?
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drrlf say: hjzvbgtg
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zjnpb say: bpcvtgbz
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L. E. Gant say: 36
==> 4*3*3

If you mean a prime factorisation for a square number, then there will always be an even number

(eg 36 = 2^2 * 3^2, which is 2*2*3*3)
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