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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, f, .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, 274f.

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.
16J 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.

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?

Operation. Explanation.

2)371 5)371 Beginning with 2 (the smallest prime

1 „,. 7A i number except the number 1), it is found

by trial not to be an exact divisor of 371. o)371 7)371 3 is not an exact divisor of 371.

123-1- 53 ;J is not an exact divisor of 371.

7 is an exact divisor of 371. Therefore 371 is a composite number, bein^ composed of 53 sevens, or of 7 fifty-threes.

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?

Properties of Numbers.

2. Is the number 397 prime or composite?

Operation. 2)397 3)397

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Explanation. By trial it is found that neither 2, 3, 5, 7,11,13, 17, nor 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 o'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 male 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.

Properties of Numbers.
152. Prime Factors.

1. An integral factor that is a prime number is a prime factor.

5 is a prime factor of 30.

7 is a prime factor of and .

3 is a prime factor of and —.

2 and 3 are prime factors of and and .

3 and 5 are prime factors of and and

2. Resolve 105 into its prime factors.

Operation. Explanation.

5)105 Since the prime number 5 is an exact divisor of 105,

owl 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 = 5x3x7. 18 = 3x3x2.

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?

Properties of Numbers.

153. Multiples, Common Multiples, And Least Common Multiples.

1. A multiple of a number is an integral number of times the number.

30 and 35 and 40 are multiples of 5.
16 and 20 and 32 are multiples of 4.

2. A common multiple of two or more numbers is an integral number of times each of the numbers.

30 is a common multiple of 5 and 3.
40 is a common multiple of 8 and 10.

is a common multiple of 9 and 6.

is a common multiple of 8 and 12.

3. A common multiple of two or more integral numbers contains all the prime factors found in every one of the numbers, and may contain other prime factors.

48 = 2x2x2x2x3. 150 = 2x3x5x5. A common multiple of 48 and 150 must contain four 2's, one 3, and two 5's. It may contain other factors.

2x2x2x2x3x5x5 = 1200. 2x2x2x2x3x5x5x2= 2400. 1200 and 2400 are common multiples of 48 and 150.

4. The least common multiple (l. c. m.) of two or more numbers is the least number that is an integral number of times each of the numbers.

40 and 80 and 120 are common multiples of 8 and 10; but 40 is the least common multiple of 8 and 10.

5. The least common multiple of two or more numbers contains all the prime factors found in every one of the numbers, and no other prime factors.

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