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Given the prime factorization of an integer how can you determine if our integer is a perfect square?

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You can't. You can only find the prime factorization of an integer.

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The positive integer factors of 293 are:

1, 293

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Negative numbers don't have prime factorizations.

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== Will you please answer my question?! Will you please answer my question?! == In number theory ( http://www.answers.com/topic/number-theory ), integer factorization is the process of breaking down a composite number ( http://www.answers.com/topic/composite-number ) into smaller non-trivial integers ( http://www.answers.com/topic/divisor-2 ), which when multiplied together equal the original integer. Source:integer-factorization

== The prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder.

The process of finding these numbers is called integer factorization, or prime factorization. Source:http://en.wikipedia.org/wiki/Prime_factor

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2x5x5x3

you can apply the negative to any of the numbers

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Expressing a positive composite integer as the product of its prime factors.

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The square root of 512 is neither an integer, nor even a rational number,

so it has no prime factorization.

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The smallest positive integer that has exactly 6 factors is 12. To determine the number of factors an integer has, we can use its prime factorization. In the case of 12, its prime factorization is 2^2 * 3, which means it has (2+1)(1+1) = 6 factors.

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The least positive integer that has exactly eight factors is 24. The number of factors of an integer can be determined from its prime factorization. The prime factorization of 24 is (2^3 \times 3^1), and using the formula for the number of factors, ((e_1 + 1)(e_2 + 1)), where (e_i) are the exponents in the factorization, we have ((3 + 1)(1 + 1) = 8). Thus, 24 has exactly eight factors.

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Shor's Algorithm is used in quantum computers and is used for integer factorization.

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This is because a factor is defined in terms of multiplication, not addition. One integer, p, is a factor of another integer, q, if there is some integer, r (which is not equal to 1) such that p*r = q.

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I would assume -1, which is not prime and must be part of the factorization to make the negative number negative.

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Since the square root of 18 is not an integer, it doesn't have a prime factorization.

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The positive integer factors of 721 are:

1, 7, 103, 721

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A prime factorization is the unique way to list any integer greater than 1 as a product of prime numbers.

An example of the prime factorization of a composite number is 20=2*2*5 or 20=2^2*5.

The prime factorization of a prime number is itself. e.g. 11=11.

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Thorkil Naur has written:

'Integer factorization' -- subject(s): Factorization (Mathematics)

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1, -1, 2, -2, 4, -4, 8, -8, 16, -16, 32, -32

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39 is simply a number. It is an integer, a whole number, and a natural number. The prime factorization of 39 is 3*13. You can go on from there.

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-32 = -25
The only prime factor of -32 is 2. Negative numbers can't have prime factorizations.

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The best integer of 23800 is simply 23800 itself, as it is already an integer. If you're looking for something specific like its factors or properties, the prime factorization of 23800 is 2² × 5² × 7 × 11, which can be useful in various mathematical contexts. However, without further context, 23800 remains the best integer representation.

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Each composite integer has one unique prime factorization, traditionally listed least to greatest.

2 x 2 x 3 x 3 = 36

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Not a lot of room between 29 and 30.

Prime factorization is the expression of a positive composite integer as the product of its prime factors.

30 has three prime factors.

2 x 3 x 5 is the prime factorization of 30.

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suppose the n has the prime factorization of x*y. We know that every unique integer has a unique prime factorization. n*n = (x*y)*(x*y) = x^2*y^2.

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Because the Fundamental Theorem of Arithmetic specifies that every integer greater than 1 has its own unique prime factorization, it is impossible to specify what each of these prime factorizations is, however, it is true that the prime factorization of every even number includes the number 2 as the lowest prime factor.

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Unique factorization usually means that any integer can only be factored in one way using prime numbers only:

24 = 2 x 2 x 2 x 3 (unique prime factorization)

If other numbers than prime numbers are allowed, factorization is not unique.

24 = 2 x 12 = 3 x 8 = 4 x 6 = -4 x -6 = etc. (non-unique factorization)

If 1 is allowed, then every number has an infinity of factorizations:

5 = 1 x 5 = 1 x 1 x 5 = 1 x 1 x 1 x 5 = etc.

So, limiting the allowed factors to prime numbers, makes the factorization unique.

The theorem is that every integer has a unique prime factorization. So, the answer to your question could be any number showing its unique prime factorization.

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In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 is either prime itself or is the product of prime numbers, and that, although the order of the primes in the second case is arbitrary, the primes themselves are not.

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Like any other integer, 22 expressed as an exponent is 22^1. The prime factorization of 22 is 2 x 11. No exponents are required.

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The positive integer factors of 2476 are:

1, 2, 4, 619, 1238, 2476

The prime factorization of 2476 is:

2476 = 22 x 619

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15432 is an integer number which is even and one bigger than 15431 and one smaller than 15433

Its Prime factorization is 2 cubed times 3 times 643.

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The positive integer factors of 644 are 1, 2, 4, 8, 83, 166, 332, 664.

The prime factorization of 644 is 23 x 83

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The positive integer factors of 440 are as follows:

1, 2, 4, 5, 8, 10, 11, 20, 22, 40, 44, 55, 88, 110, 220, 440

The prime factorization of 440 is as follows:

23 x 5 x 11 = 440

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Thanks to the fundamental theorem of arithmetic, there can be only one prime factorisation (leaving aside the factor 1), for ANY integer. As a result, there is only one prime factorisation of 30 and so nothing for it to be different from!

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Yes, 42 and 77 are relatively prime because they have no common positive integer factors other than 1. The prime factorization of 42 is 2 × 3 × 7, while the prime factorization of 77 is 7 × 11. The only common factor is 7, so they are not relatively prime.

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When asked for a factorization, it is generally understood to mean the prime factorization.

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It's a question of how "prime" is defined. Mathematicians have chosen not to include 1 in their set of prime numbers, possibly because they feel it makes things easier.

Every positive integer can be written as a product of prime numbers (prime factorization). In fact, every positive integer has only one prime factorization. (Reordering the primes doesn't count, e.g. 2x3x3 is the same as 3x2x3.) This would no longer be true if 1 was a prime number. For instance: 14 = 2x7 = 1x2x7 = 1x1x2x7 = 1x1x1x2x7. Every positive integer would have infinitely many prime factorizations. You's have to rewrite the theorem to say "...only one prime factorization WITHOUT A 1", which would be more awkward.

There are probably more examples out there, but I can't think of any good ones at the moment.

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No, it is an integer.

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The Highest Common Factor (HCF) of 120 and 65 is the largest positive integer that divides both 120 and 65 without leaving a remainder. To find the HCF, you can use the Euclidean algorithm or prime factorization method. In this case, the prime factorization of 120 is 2^3 * 3 * 5 and the prime factorization of 65 is 5 * 13. The common factors are 5, so the HCF of 120 and 65 is 5.

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The absoluate value of a positive integer is the integer itself.

The absoluate value of a positive integer is the integer itself.

The absoluate value of a positive integer is the integer itself.

The absoluate value of a positive integer is the integer itself.

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Prime factorization of positive integers is a list of the integer's prime factors. The product of the common prime factors of 48 is 2 x 2 x 2 x 2 x 3.

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Yes, the square of an integer is always an integer.

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The square root of an integer is a CYCLOTOMIC integer.

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1448 is an integer.

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Factorization of 150 = 2x3x5x5.

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Factorization into its smallest factors.

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