101. A Prime Factor is a factor that is a prime number. 102. A Composite Number is equal to the product of all its prime factors. Thus,

8 = 2 × 2 × 2, and 12 = 2 × 2 × 3.

103. The number of times a number is taken as a factor, is sometimes denoted by a small figure, called an exponent, written at the right of the figures expressing the factor, and above the line. Thus.

282 X2 X 2, and 1832 X 2.

104. Two or more numbers are said to be prime with respect to each other, when they have no common integral factor, except Thus,

1.

4 and 9 are prime with respect to each other.

105. Factoring is the process of finding the factors of composite number.

EXACT DIVISORS.

106. Two is an exact divisor of every even number, but of no odd number. Thus,

2 is an exact divisor of 2, 4, 6, etc., but not of 3, 5, 7, etc.

107. Three is an exact divisor of any number, when it is an exact divisor of the sum of the units expressed by its figures. Thus,

3 is an exact divisor of 546.

108. Four is an exact divisor of any number, when it is an exact divisor of its tens and units. Thus,

4 is an exact divisor of 572, 1928.

109. Five is an exact divisor of any number whose unit figure is 5 or 0. Thus,

5 is an exact divisor of 15, 20, 25, 30.

A Prime Factor? To what is a composite number equal? When are two or more numbers said to be prime to each other? What is factoring? Of what numbers is two an exact divisor? Three? Four? Five?

110.

Six is an exact divisor of any even number of which is an exact divisor.

Thus,

6 is an exact divisor of 12, 18, 24, 30.

111. Nine is an exact divisor of any number, when it is an exact divisor of the sum of the units expressed by its figures. Thus,

9 is an exact divisor of 7542, as 9 is an exact divisor of 7+ 5+4+2=18.

PRIME NUMBERS.

112. No direct method of detecting prime numbers has been discovered.

After 2, there can be no even number prime, since 2 is an exact divisor of every even number (Art. 106).

In a series of odd numbers beginning with 1, it has been found by trial, that every third number after the prime 3 has 3 as a factor, every fifth number after the prime 5 has 5 as a factor, and so on.

113. Hence, we have a practical method of detecting prime numbers by sifting out those that are not prime, as follows:

Write the odd numbers from 1 to any desirable limit.

Begin with the first prime number after 2, which is 3, and mark every third number after the 3 by writing 3 over it, every fifth number after the 5 by writing 5 over it, every seventh number after the 7 by writing 7 over it, and so on.

Then, all that remain are prime numbers; and those marked are composite, with factors over them. Thus,

3, 5,
5, 7,

3

3, 5

3,7

5

9, 11, 13, 15, 17, 19, 21, 23, 25,

1, 3,

Six? Nine? What is said with regard to detecting prime numbers? What is the only even prime number? Give the practical method of detecting prime numbers.

114. Prime numbers to 1009 are included in the following

Table of Prime Numbers.

53 137 229 331 433 547 647 761 881 1009

Every prime number, except 2 and 5, has 1, 3, 7, or 9 for its unit figure.

FACTORING OF NUMBERS.

115. To resolve or separate a nuniber into its prime factors.

1. Resolve or separate 84 into its prime factors.

Ans. 842 X2 X3 X 7 = 22 × 3 × 7.

What has every prime number, except 2 and 5, for its unit figure?

By trial, we find that 84 is composed of two factors, 2 and 42; of which 2 is prime and 42 is composite.

The 42 we find composed of two factors, 2 and 21; of which 2 is prime and 21 is composite.

The 21 is composed of two factors, 3 and 7, both prime.

Therefore, the prime factors of 84 are 2, 2, 3, and 7, and may written 22, 3, and 7.

be

RULE. Divide the given number by any prime number greater than 1, that is an exact divisor, and the quotient, if composite, in the same manner; and thus continue until the quotient is prime. The divisors and the last quotient will be the prime factors required.

PROOF. The product of the prime factors will equal the given number, if the work is right (Art. 102).

Since 1 is a factor of every number, it is not commonly specified as such.

116. To find the prime factors common to two or more numbers, we may

Take out the common factors, by dividing the given numbers by any prime number greater than 1, that is an exact divisor of them all, and treat the quotients in the same manner, until quotients shall be obtained prime to each other.

12. What are the prime factors common to 28 and 56?

What is the Rule? The Proof? How may the prime factors common to two or more numbers be found?

OPERATION.

2 28, 56 214, 28 7 7, 14

2

Ans. 2, 2, and 7, or 22 and 7.

By trial, we find 2 to be an exact divisor of all the numbers, and we therefore take it as one of the common factors, and have left of 28 the factor 14, and of 56 the factor 28.

We find 2 to be an exact divisor of the factors 14 and 28, and we

take it as another common factor of the given numbers, and have left of 28 the factor 7, and of 56 the factor 14.

We find 7 is an exact divisor of the factors 7 and 14, and we take it as a common factor of the given number, and we have left only a factor 2 of 56, and which cannot be common to 28. are the common prime factors of 28 and 56.

What are the prime factors common,

13. To 45 and 75.

14. To 99, 165, and 330.

15. To 60, 210, and 390.

Therefore, 2, 2, and 7

Ans. 3 and 11. Ans. 2, 3, and 5.

MULTIPLICATION BY FACTORS.

117. 1. Multiply 452 by 35, using factors.

OPERATION.

452

7

3164

5

Ans. 15820

35 is equal to 7 times 5; hence 35 times 452 is equal to 5 times 7 times 452.

7 times 452 is equal to 3164, and 5 times 7 times 452, or 5 times 3164, is equal to 15820.

Therefore, 452 multiplied by 35 is equal to 15820. Had 35 contained any other convenient set of factors, they could have been used in like manner.

RULE. Separate the multiplier into convenient factors. Multiply the multiplicand by one of these factors, and the product by another, and so on, until all the factors have been used. The last product will be the one required.

Repeat the Rule.