Prime factorization is a fundamental concept in mathematics, essential for understanding the building blocks of numbers. This article delves into the definition of prime factors, the process of prime factorization, and provides step-by-step examples to illustrate the method. We also explore the Fundamental Theorem of Arithmetic and the application of divisibility rules.
Prime factors are the prime numbers that multiply together to give a natural number. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.
The multiplicity of a prime factor is the number of times it appears in the prime factorization of a number.
Prime factorization is the process of expressing a natural number as a product of its prime factors. According to the Fundamental Theorem of Arithmetic, every natural number greater than 1 has a unique prime factorization, except for the order of the factors.
2 | 144
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2 | 72
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2 | 36
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2 | 18
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3 | 9
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3 | 3
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end | 1
Prime factorization of 144: (144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 = 2^4 \times 3^2).
2 | 420
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2 | 210
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3 | 105
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5 | 35
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7 | 7
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end | 1
Prime factorization of 420: (420 = 2 \times 2 \times 3 \times 5 \times 7 = 2^2 \times 3 \times 5 \times 7).
Check divisibility by 2, 3, 5: Not divisible.
Check divisibility by 7:
Check divisibility by 11:
Check divisibility by 13:
7 | 17017
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11 | 2431
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13 | 221
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17 | 17
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end | 1
Prime factorization of 17017: (17017 = 7 \times 11 \times 13 \times 17).
For more detailed information on prime factorization, visit Math is Fun.
Prime factorization is a crucial mathematical process that breaks down numbers into their prime components. Understanding this concept is essential for various applications in number theory and cryptography. By following the steps outlined and using divisibility rules, you can efficiently find the prime factors of any natural number.
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