PROPERTIES OF NUMBERS. To the Teacher: Under this head, number in the abstract is discussed with little or no distinction between numbers of things and pure number. It is dissociation and generalization without which there could be little progress in the "science of number" or in the "art of computation." 146. Every number is fractional, integral,or mixed. 1. A fractional number is a number of the equal parts of some quantity considered as a unit; as, }, .9, 5 sixths. 2. An integral number is a number that is not, either wholly or in part, a fractional number; as, 15, 46, ninetyfive. 3. A mixed number is a number one part of which is integral and the other part fractional; as, 5, 27.6, 274§. 147. An exact divisor * of a number is a number that is contained in the number an integral number of times. 5 is an exact divisor of 15. 5 is not an exact divisor of 1.5. 163 is an exact divisor of 100. 148. Every integral number is odd or even. 1. An odd number is a number of which two is not an exact divisor; as, 7, 23, 141. 2. An even number is a number of which two is an exact divisor; as, 8, 24, 142. *Note.-An exact divisor of a number is sometimes called an aliquot part of the number. Properties of Numbers. 149. Every integral number is prime or composite. 1. A prime number is an integral number that has no exact integral divisors except itself and 1; as, 23, 29, 31, etc. 2. Is two a prime number? three? nine? 3. Name the prime numbers from 1 to 97 inclusive*. Find their sum. 4. A composite number is an integral number that has one or more integral divisors besides itself and 1; as, 6, 8, 9, 10, 12, 14, 15, etc. 5. Name the composite numbers from 4 to 100 inclusive. Find their sum. 6. Is eight a composite number? eleven? fifteen? (a) Find the sum of the results of problems No. 3 and No. 5. 150. To find whether an integral number is prime or composite. 1. Is the number 371 prime or composite? 2)371 Operation. 185+ 74+ 3)371 7)371 123+ 53 Explanation. Beginning with 2 (the smallest prime number except the number 1) it is found by trial not to be an exact divisor of 371. 3 is not an exact divisor of 371. 5 is not an exact divisor of 371. 7 is an exact divisor of 371. Therefore 371 is a composite number, being composed of 53 sevens, or of 7 fiftythrees. t Observe that we use as trial divisors only prime numbers. If 2 is not an exact divisor of a number, neither 4 nor 6 can be. Do you see why? *There are 26 prime numbers less than 100. + Note the similarity of the words composed and composite. Properties of Numbers. 2. Is the number 397 prime or composite? Operation. 2)397 198+ 132+ 5)397 7)397 79+ 56+ 11)397 13)397 36+ 17)397 30+ 19)397 23+ 20+ Explanation. By trial it is found that neither 2, 3, 5, 7, 11, 13, 17 or 19 is an exact divisor of 397. No composite number between 2 and 19 can be an exact divisor of 397: for since one 2 is not an exact divisor of the number, several 2's, as 4, 6, 8, 12, etc., cannot be; since one 3 is not an exact divisor of the number, several 3's, as 6, 9, 12, etc., cannot be; since one 5 is not an exact divisor of the number, several 5's, as 10 and 15 cannot be; since one 7 is not an exact divisor of the number, two 7's (14) cannot be. No number greater than 19 can be an exact divisor of the number; for if a number greater than 19 were an exact divisor of the number the quotient (which also must be an exact divisor) would be less than 20. But it has already been proved that no integral number less than 20 is an exact divisor of 397. Therefore 397 is a prime number. Observe that in testing a number to determine whether it is prime or composite, we take as trial divisors, prime numbers only, beginning with the number two. Observe that as the divisors become greater, the quotients become less, and that we need make no trial by which a quotient will be produced that is less than the divisor. 3. Determine by a process similar to the foregoing, whether each of the following is prime or composite: 127, 249, 257, 371. 151. Any divisor* of a number may be regarded as a factor of the number. An exact integral divisor of a number is an integral factor of the number. * NOTE.-The word factor is often loosely used for integral factor. Properties of Numbers. 152. PRIME FACTORS. 1. An integral factor that is a prime number is a prime 5)105 3)21 7 Explanation. Since the prime number 5 is an exact divisor of 105, it is a prime factor of 105. Since the prime number 3 is an exact divisor of the quotient (21), it is a prime factor of 21 and 105. Since 3 is contained in 21 exactly 7 times, and since 7 is a prime number, 7 is a prime factor of 21 and of 105. Therefore the prime factors of 105 are 5, 3, and 7. Observe that if 7 and 3 are prime factors of 21 they must be prime factors of 105, for 105 is made up of 5 21's. 7 is contained 5 times as many times in 105 as it is in 21. Observe that every composite number is equal to the product of its prime factors. 105 5 X 3 X 7. 18 = 3 x 3 x 2. = 3. Resolve each of the following numbers into its prime factors: 224, 15, 21, 6. Then prove that the continued product of the numbers is equal to the continued product of all their factors. Observe that 2 times 3 times a number equals 6 times the number; 3 times 5 times a number equals 15 times the number, etc. Observe that instead of multiplying a number by 21, it may be multiplied by 3 and the product thus obtained by 7, and the same result be obtained as would be obtained by multiplying the number by 21. Why? |