How to Find the Largest Prime Factor of a Composite Number
The relationship between a composite -- a positive integer that is not a prime number -- and prime numbers is that the former can be expressed as a product of latter. A prime number, by definition, is one that is divisible only by itself and 1 (by convention, 1 is not considered a prime number), and a composite number is one that has a divisor other than itself and 1. The fundamental theorem of arithmetic states that any positive number is either a prime number or can be uniquely factored into prime numbers.
Before factoring a composite into its primes, it is necessary to have at hand a list of small primes. The first 10 primes are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. See Resources for lists of primes.
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Instructions
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Divide your number by the smallest prime number, 2. If it is not divisible by 2, try the next prime, 3. Continue until you find the lowest prime factor for your number. For example, for 42, the first prime divisor is 2. Thus, it can be expressed as a product of 2 * 21.
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Repeat Step 1 for the largest remaining number in the product. In the example, the next prime divisor is 3. The expression now becomes 2 * 3 * 7. Notice that all the factorization is complete because all the remaining factors are prime numbers.
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Continue to factor with lowest possible prime until you are left with prime factors only.
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Select the largest number from the product of factors. This is the largest prime factor of your composite number. In the example, 42 is factored into 2 * 3 * 7. Thus, the highest prime factor for 42 is 7.
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Consult the "Find the Prime Factors of a Number" Web page (see Resources) and enter your number. Calculating prime factors for large numbers is time-consuming. Specialized software can make the job fast and easy.
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References
Resources
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