Machinery's Handbook, 31st Edition
14 PRIME FACTORIZATION OF NUMBERS 4 is a factor of any number in which the last two digits are a number divisible by 4. Thus, 1844 has a factor 4, because 44 ÷ 4 = 11. 761 does not have a factor of 4, since 61 is not divisible by 4. 5 is a factor of any number that has a ones digit that is either 0 or 5. A prime number is one that has no factors except itself and 1. Thus, 2, 3, 5, 7, 11, etc. are prime numbers. 2 is the only even prime number. A factor which itself is a prime number is called a prime factor . All numbers can be expressed as a product of their prime factors. It can be determined if 7 is a factor of a number according to this process: Remove the last digit from the number, double it, and subtract it from the remaining number. If the result is divisible by 7 (e.g., 14, 7, 0, –7, etc.), then the number is divisible by 7. The prime factorization of a number is done by expressing the number as a product of its primes. For example, the prime factors of 20 are 2 and 5; the prime factorization is 2 × 2 × 5 = 20. The Prime Number and Factor Table , starting on page 15, give the smallest prime factor of all odd numbers from 1 to 9600, and can be used for finding all the factors for numbers up to this odd number. Where no factor is given for a number in the table, the letter P indicates that the number is a prime number. The last page of the tables lists prime numbers from 9551 through 18691; it can be used to identify unfactorable numbers in that range. Example 1: Find the factors of 833. Use the table on page 15 as illustrated below. Solution: The table on page 15 indicates that 7 is the smallest prime factor of 833, shown at the row-column intersection for 833. This leaves another factor, because 833 ÷ 7 = 119. From To 0 100 100 200 200 300 300 400 400 500 500 600 600 700 700 800 8 00 900 900 1000 1000 1100 1100 1200
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It also shows that 7 is a prime factor of 119, leaving a factor 119 ÷ 7 = 17. From To 0 100 100 200 200 300 300 400 400 500 500 600 600 700 700 800 800 900 900 1000
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P indicates that 17 is a prime number and no other prime factors of 833 exist. From To 0 100 100 200 200 300 300 400 400 500 500 600 600 700 700 800 800 900 900 1000 1000 1100
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Hence, the prime factorization of 833 is 7 × 7 × 17. Example 2: A set of four gears is required in a mechanical design to provide an overall gear ratio of 4104 ÷ 1200. Furthermore, no gear in the set is to have more than 120 teeth or less than 24 teeth. Determine the tooth numbers. Solution: The prime factorization of 4104 is determined to be 2 × 2 × 2 × 3 × 3 × 57 = 4104. The prime factorization of 1200 is determined to be 2 × 2 × 2 × 2 × 5 × 5 × 3 = 1200. Therefore, 4104 . Each resulting factor represents the number of teeth that fulfill the requirement. If the factors had been com bined differently, say, to give 72 57 × 16 75 × --------- , then the 16-tooth gear in the denominator would not satisfy the requirement of having no less than 24 teeth. Example 3: Factor 25,078 into two numbers, neither of which is larger than 200. 1200 ------ 2 2 × 2 × 3 × 3 × 57 × 2 2 × 2 × 2 × 5 × 5 × 3 × ----------------------------- 72 57 × 24 50 × --------- = = Solution: The smallest factor of 25,078 is obviously 2, leaving 25,078 ÷ 2 = 12,539 to be factored further. However, from the last table, Prime Numbers from 9551 to 18691 , on page 23, it is seen that 12,539 is a prime number; therefore, no other factors exist. So the factorization named is not possible.
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