It is wrong to say that the prime factors of 24 are 2 times 2 times 2 times 3, as many do. Accuracy in statement means much, and carelessness is never more out of place than in arithmetic. 40 = 4 × 10 40 = 5 x 8 5} Show the child that These are factors of 40, but not all of them are prime factors. 40 can have only one set of prime factors. while many numbers have two or more sets of factors, there is only one set of prime factors. 24 = 2 × 2 × 2 × 3 27 = 3 × 3 × 3 25 = 5 × 5 In factoring numbers many times we find the same factor in two or more of them. The factor 3 is found in 24, also in 27. Then 3 is a common factor of 24 and 27. When we find no factor common to two or more numbers, we say the factors are prime to each other. The factors of 27 and 25 are prime to each other. Are the factors of 25 and 24 prime to each other? FACTORING SMALL NUMBERS. Small numbers are factored by inspection. The child should be drilled until he can name a prime factor of any small number that may be named. Give him the number 14. He sees at a glance that a prime factor of 14 is 2. Give him the number 135. He does not see without drill that 5 is a prime factor, also 3. The thing to do is to drill on the principles given here, that the child may quickly tell if a prime factor is 2, 3, 5, 7, etc. How to Tell One Prime Factor of a Number: 1. One factor is 2, if the number ends in 0, 2, 4, 6, 8. 2. One factor is 3, if 3 is a factor of the sum of the digits. 3. One factor is 5, if the number ends in 0 or 5. 4. One factor is 7, if the difference between the sum of the odd periods and the sum of the even periods is divisible by 7. NOTE. It requires about as much time to apply this last test as it does to divide by 7, so it is seldom used. The others are practical. FACTORING LARGE NUMBERS. There are two ways of factoring large numbers, both of them being used. Things for the Child to Do: 1. Name the six factors of 24. 2. Is there any prime number larger than 2 that is not an odd number? 3. Find all the prime numbers from 3 to 63. 4. Name all the composite numbers from 28 to 60. Prime Numbers from 1 to 1000. PRIME NUMBER TABLE. PRESENT IN THIS WAY: What number will exactly divide 12 and 18? (Child) 2. 2 is then a divisor of 12 common divisor of 12 and 18. common factor of 12 and 18. It is a and of 18. Is 2 the only common divisor of 12 and 18? (Child) No, 3 and 6 are also common divisors of 12 and 18. What is the Greatest Common Divisor of 12 and 18! (Child) 6. Who can tell what is meant by the Greatest Common Divisor of two or more numbers? Name two common divisors of 10 and 20. What is the Greatest Common Divisor of 28 and 64 ? The Greatest Common Divisor (G. C. D.) of two or more numbers is the largest number contained in each of them a whole number of times. NOTE.- Greatest Common Measure and Greatest Common Factor mean the same as Greatest Common Divisor and are used in some schools. ILLUSTRATION: Find the G. C. D. of 12, 18, 36. The divisors of 18 are 2, 3, 6, 9. The divisors of 36 are 2, 3, 4, 6, 9, 12, 18. 2 is a common divisor of 12, 18, and 36, because it is found in each of the numbers as a factor; but it is not the Greatest Common Divisor. 6 is the G. C. D. of 12, 18, 36. Review: 1. Review the definitions of prime and composite factors. 2. Drill the child to recognize prime factors of numbers. 3. Have him name all the prime divisors of 42; of 50; of 30; of 36. 4. Have him name all the divisors of the above numbers. 5. Name all their factors. 6. What is the difference between a factor and a divisor of a number? 7. Can there be more than one G. C. D. of a set of numbers? Remember: 1. Every factor of a number is a divisor of that number. 2. Any number can be divided by the product of two or more of its prime factors. FINDING THE G. C. D. OF SMALL NUMBERS. We can find the G. C. D. of small numbers by writing the common divisors of the numbers and then selecting the greatest one, as we did in the first part of this lesson. But there is a better way. ILLUSTRATION: Find the G. C. D. of 12, 18, and 36. 2 and 3 are common prime factors, or divisors, of 12, 18, and 36. Multiply the common prime factors together and the result is the Greatest Common Divisor. 2 × 3 = 6. 6 is the G. C. D. of 12, 18, and 36. Principle. The Greatest Common Divisor of two or more numbers is :he product of their common prime factors. ILLUSTRATION: Find the G. C. D. of 24, 48, and 72. |
