FACTORING 



2124 



FACTORING 



these three numbers has other factors; for 

 example, 2 and 21 are factors of 42; 2 and 9 

 are factors of 18; 3, 8 and 3 are factors of 72. 



A number that has no factors except itself 

 and 1 is a prime number, as 19, 3, 37, 2, 101. 



A prime number used as a factor is a prime 

 factor; for example, in 3X7=21, 3 and 7 are 

 T>rime factors of 21; in 5X7X23=805, 5, 7 and 

 23 are prime factors of 805. 



A number that has factors besides itself and 

 1 is a composite number; for example, 4, 9. 

 25, 150. 



A composite number used as a factor is a 

 composite factor; as, 4X9=36. 4 and 9 are 

 composite factors of 36. 4X25=100. 4 and 25 

 are composite factors of 100. 



A factor of a number will divide the number 

 exactly (without a remainder) and so is called 

 a divisor of the number. 3X7=21. 3 and 7 

 are factors of 21. 21^3=7; 21-^7=3. They 

 are also divisors of it. 



Exercises. (1) Name all the prime numbers 

 from 1 to 100. How can you tell which of 

 these numbers are prime? (See DIVISIBILITY OF 

 NUMBERS, and study it in connection with fac- 

 toring.) 



(2) Using the "tests for divisibility," tell by 

 what numbers from 2 to 11, inclusive, each of 

 the following is divisible: 



84 384 402 



81 561 1168 



27 9729 7986 



75 37800 105 



(3) Tell which of the following numbers are 

 prime and which composite: 



26 441 88 3069 



18 79 41 671 



59 111 171 407 



38 119 67 729 



66 167 57 1023 



Separation of a Number into Prime Factors. 

 (1) Let us separate 72 into its prime factors. 



(a) 72 = 8X9 



(b) 72=(2X2X2)X(3X3) 



(c) 72 = 2X2X2X3X3 



In (a) 72 is separated into two composite 

 factors, 8 and 9. In (b) these factors are 

 separated into their prime factors, and we 

 have the prime factors of 72. In (c) the 

 parentheses are dropped; (c) may be written 

 in another form, which is very convenient, by 

 using the exponent, thus: 72=2 3 X3 2 . The 

 exponent 3 tells how many times 2 is used as 

 a factor in 72, and the exponent 2 tells how 

 many times' 3 is used as a factor in 72. 



(2) We shall factor 81, showing the steps as 

 above. 



81 = 9* 

 81 = 9X9 



81=(3X3)X(3X3) 

 81=3X3X3X3 

 81 = 3 



The exponent 4 shows that 3 is used as a factor 

 4 times in 81. 



(3) Factor 144. 



144 = 12* 



144 = 2X6X2X6 



144 = 2X3X2X2X3X2 



144 = 2X3 J 



Another method of finding the prime factors 

 of a number follows: 



(4) Find the prime factors of 1365. 1365 is 

 divisible by 5, since it ends in 5. Divide, and 

 we see that 1365=5X273. 273 is divisible by 

 3, since the sum of its digits is divisible by 3. 

 Divide, and we find that 1365=5X3X91. Di- 

 viding 91 by 7, we find 1365=5X3X7X13. A 

 more concise form is this: 



These steps may be summarized as 

 follows : 



511365 



31273 



7191 



13 (1) Separate the number Into any of 

 its composite factors and then separate these fac- 

 tors into their prime factors, as in (1), (2) and 

 (3) above. 



(2) Divide the number by one of its prime 

 factors ; then divide the quotient obtained by one 

 of its prime factors ; continue this process until 

 the quotient is a prime number. The divisors 

 and the last quotient are the prime factors of the 

 number, as in (4) above. 



Common Factor. A factor that occurs in 

 each of two or more numbers is called a 

 common factor of the several numbers; for 

 example, 8 is a factor common to 16, 24, 48, 72 ; 

 7 is common to 28, 56, 105. A common factor 

 is called also a common divisor and a common 

 measure, because it divides each number ex- 

 actly, and with it as a measure we may 

 evaluate and compare the several numbers; for 

 example, 7 is a common measure of 35, 42, 84 

 and 21. 35 is 5 sevens, 42 is 6 sevens, 84 is 

 12 sevens and 21 is 3 sevens. 



Greatest Common Factor. The largest factor 

 or divisor which is found in each of two or 

 more numbers is called the greatest common 

 factor, the greatest common divisor, and the 

 greatest common measure of the several num- 

 bers. For example, 24 is the largest factor 

 found in 48, 72 and 96. Therefore 24 is the 

 greatest common factor, the greatest common 

 divisor and the greatest common measure of 

 48, 72 and 96. 



Numbers Prime to Each Other. 16 and 45 

 have no common factor other than 1 ; 25 and 





