Hence, to separate a composite number into its prime factors, Divide the given number by any prime number that is contained in it without remainder, then the quotient by any prime number that is contained in it without remainder, and so on until the quotient is itself a prime number. The several divisors and the last quotient are the prime factors required. 222. The following tests are very useful for determining without actual division whether a number contains certain factors: 1. A number is divisible by 2 if its last digit is even. 2. A number is divisible by 4 (22) if the number denoted by the last two digits is divisible by 4. 3. A number is divisible by 8 (23) if the number denoted by the last three digits is divisible by 8. 4. A number is divisible by 3 if the sum of its digits is divisible by 3. 5. A number is divisible by 9 (32) if the sum of its digits is divisible by 9. 6. A number is divisible by 5 if its last digit is either 5 or 0. 7. A number is divisible by 25 (52) if the number denoted by the last two digits is divisible by 25. 8. A number is divisible by 125 (53) if the number denoted by the last three digits is divisible by 125. 9. A number is divisible by 6 if its last digit is even and the sum of its digits is divisible by 3. 10. A number is divisible by 11 if the difference between the sum of the digits in the even places and the sum of the digits in the odd places is either O or a multiple of 11. NOTE. The shortest method of dividing by 25 is to multiply by 4 and divide by 100; by 125, is to multiply by 8 and divide by 1000. In adding the digits of a number to determine whether their sum is a multiple of a certain number, omit those digits which are seen at a glance to be multiples of the number. Thus, to discover whether 8,983,167 is divisible by 3, omit 9, 3, 6 (8, 1), which are manifestly multiples of 3, and simply add 8 and 7. 223. Other prime factors, 7, 13, 17, 19, sometimes betray their presence to one familiar with the subject; but, practically, the best way to detect them is to attempt to divide by them. 224. If we divide any number less than 121 (112) by 11, or by a number greater than 11, it is plain that the quotient is less than 11. If we divide any number between 121 and 143 (11 × 13) by 11, the quotient will evidently lie between 11 and 13; and, since there are no prime numbers between 11 and 13, the quotient, if a whole number, must be composite, and contain factors smaller than 11. What is thus proved of 11 and 13 is evidently true of any two adjacent prime numbers; namely, that, excepting the second power of the smaller prime number, every composite number less than the product of two adjacent prime numbers, contains prime factors less than the smaller of these two numbers. Thus, every composite number less than 4087 (61 x 67), except 3721 (612), contains prime factors less than 61. 225. From the preceding article, the value of the following table, in discovering the prime factors of a given. number, will be apparent. 101 103 107 109 113 Primes. 79 83 89 97 Opposite to "Powers” are placed the squares of the primes from 7 to 109; and opposite to "Products" are placed the products of the successive pairs of adjacent primes from 7 to 113. 226. Find the prime factors of 610,764. As 64 is divisible by 4, but 764 is not divisible 22 610,764 by 8, 22 is the highest power of 2 contained in 3 152,691 7 50,897 11 7,271 610,764. As the sum of the digits 152,691 is divisible by 3 but not by 9, 31 is the highest power of 3 contained in 152,691. The next quotient, 50,897, does not contain 5; but divided by 7 gives 7271. 7271 does not contain 7; but, since 7+ 7 — (2 + 1) = 11, it is divisible by 11. The quotient 661 when divided by 6 gives a remainder of 1, which shows that it may be a prime number. It cannot be divided by 11, 13, 17, or 19, and is seen by the table to be less than 667 (23 × 29), and not equal to 529 (232); therefore it is a prime number. Thus, 610,764 22 × 3 × 7 × 11 × 661. =3 EXERCISE XI. Find the prime factors of: 1. 148; 264; 178; 183; 173; 187; 346; 343; 227. A number is not only divisible by each of its prime factors, but by every possible combination of them. For example, 120 is 23 × 3 × 5, and is divisible either by 2, 4, 8, 6, 12, 24, 30, 60, 10, 20, 40, or 15. But 228. The number 14.21 may be put in the form of 1421 x .01; and be thus resolved into 72 x 29 x .01. .01 is not properly a factor, it is a divisor; it is the reciprocal of 22 × 52. Nevertheless, it is frequently of great practical advantage to separate mixed decimals, in this way, by first taking out the apparent factors .1, .01, .001, etc. Thus, the factors of 142.1 may be said to be 7, 7, 29, and .1; of 1.421, 7, 7, 29, and .001. EXERCISE XII. Find the prime factors of: 1. 8.4; 7.6; 1.08; .144; .036; .037; 21.45; 3. 8.67; 48.3; 99.99; 5.04; 1.485; .216; GREATEST COMMON MEASURE. 229. The measures of 30 and 50 respectively are: 1, 2, 3, 5, 6, 10, 15, 30; 1, 2, 5, 10, 25, 50. It will be seen that these two numbers have the measures 1, 2, 5, 10 in common, and of these measures 10 is the greatest. 230. The measures that two or more numbers have in common are called their common measures, and the greatest of these is called their Greatest Common Measure, which for the sake of brevity is indicated by the letters G. C. M. 231. If two or more numbers have no common measure except unity, they are said to be prime to each other. Thus 8 and 27 are prime to each other. 232. The prime factors of 30 are 2, 3, 5. The prime factors of 50 are 2, 52. The prime factors common to 30 and 50 are 2, 5. The G. C. M. of 20 and 30, namely 10, is 2 × 5. That is, The G. C. M. of two or more numbers consists of the prime factors common to the numbers, each prime factor having the lowest exponent that it has in any one of the numbers. 233. Hence, to find the G. C. M. of two or more numbers, Separate the numbers into their prime factors. Select the lowest power of each factor that is common to the given numbers, and find the product of these powers. Find the G. C. M. of 108, 396, 1440. |