DEF. 10. A multiple (dividend), of a number is the product arising from taking it a certain number of times: thus, 6 is a multiple of 2, because it is equal to 2 taken 3 times. Hence,

A multiple of a number can be divided by it without a remainder. Therefore, every multiple is a composite number.

DEF. 11. A factor of a number is a number that will exactly divide it: thus, 4 is a factor of 8, 12, 16, &c.

REM. The terms, factor, divisor and measure, all mean the same thing. Every composite number being the product of two or more factors, each factor must exactly divide it, (Art. 37).

Hence, every factor of a number, is a divisor of that number.

DEF. 12. A prime factor of a number is a prime number that will exactly divide it: thus, 3 is a prime factor of 12; while 4 is a factor of 12, but not a prime factor.

Therefore, all the prime factors of a number, are all the prime numbers that will exactly divide it: thus, 1, 3, and 5, are all the prime factors of 15.

Every composite number is equal to the product of all its prime factors: thus, all the prime factors of 10 are 1, 2, and 5; 1X 2X5=10.

DEF. 13. An ALIQUOT part of a number, is a number that will exactly divide it: thus, 1, 2, 3, 4, and 6, are aliquot parts of 12.

RESOLVING NUMBERS INTO PRIME FACTORS.

ART. 111. The smaller composite numbers may be resolved into their prime factors by inspection; thus,

6=2X3; 8=2×2×2; 9=3X3; 10=2X5.

In the case of large numbers, their factors are found by trial; that is, by dividing by each of the prime num

REVIEW.-110. REM. Are the even numbers prime or composite? Are the odd numbers? What is a divisor of a number? Give examples. When is one number divisible by another? Give examples.

110. What is a multiple? Give examples. A factor? Give examples. REM. What terms besides divisor are used in the same sense? Why is every factor a divisor? What is a prime factor? Give an example.

bers 2, 3, 5, 7, &c.; the prime factors of any number, being all the prime numbers that will exactly divide it.

In determining either the factors, or the prime factors, of a number, observe the following principles.

PRINCIPLE 1.-A factor of a number is a factor of any multiple of that number.

Thus, 3 is a factor of 6, and of any number of times 6; for, 6 is 2 threes, and any number of times 6 will be twice as many times 3.

PRINCIPLE 2. A factor of any two numbers is also a factor of their sum.

Since each number contains the factor a certain number of times, their sum must contain it as many times as both numbers. Thus, 2 being a factor of 6 and 8, it is a factor of their sum; for, 6 is 3 twos, and 8 is 4 twos, and their sum is 3 twos+4 twos, 7 twos.

ART. 112. From these two Principles, are derived

SIX PROPOSITIONS.

PROP. 1. Every number ending with 0, 2, 4, 6, or 8, is divisible by 2.

ILLUSTRATION.-Every number ending with 0, is either 10 or some number of tens; and, since 10 is divisible by 2, any number of tens will be divisible by 2. Prin. 1.

Again: any number ending with 2, 4, 6, or 8, may be considered a certain number of tens, plus the figure in units' place:

And, as each of the two parts of the number is divisible by 2, therefore, Prin. 2, the number itself is divisible by 2.

Conversely: No number is divisible by 2, unless it ends with a 0, 2, 4, 6, or 8.

PROP. II. Every number is divisible by 4, when the number denoted by its first two digits is divisible by 4.

REVIEW.-110. What is an aliquot part of a number? Give examples. 111. How may the smaller composite numbers be resolved into prime factors? What are the prime factors of 6? Of 8? Of 9? Of 10?

111. In determining the factors of a number, what two principles are used? Explain the first principle. The second.

ILLUSTRATION.-Since 100 is divisible by 4, any number of hundreds is divisible by 4; and any number of more than two places of figures, may be regarded as a certain number of hundreds, plus the number denoted by the first two digits.

Then, since both parts of the number are divisible by 4, Prin. 2, the number itself is divisible by 4.

Conversely: No number is divisible by 4, unless the number denoted by its first two digits is divisible by 4.

PROP. III. Every number is divisible by 5, when its right hand digit is 0 or 5.

ILLUSTRATION.-Ten being divisible by 5, and every number consisting of two or more places of figures, being composed of tens, plus the figure in the units' place:

Therefore, if this is 5, both parts of the number are divisible by 5; hence, Prin. 2, the number itself is divisible by 5.

Conversely: No number is divisible by 5, unless its right hand digit is 0 or 5.

PROP. IV. Every number whose first digits are 0, 00, &c., is divisible by 10, 100, &c.

ILLUSTRATION.-If the first figure is a cipher, the number is either 10, or some multiple of 10; and,

If the first two figures are ciphers, the number is either 100, or some multiple of 100; hence, Prin. 1, the proposition is true.

Conversely: No number is divisible by 10, 100, &c., unless it ends with 0, 00, &c.

PROP. V. Every composite number is divisible by the product of any two or more of its prime factors.

ILLUSTRATION. -Thus, the number 30 is equal to 2×3×5; now, if 30 be divided by the product of either two of the factors, the quotient must be the other factor; if not so, the product of the three factors would not be 30: and,

The same may be shown of any other composite number.

divisible by 2?

REVIEW.-112. When is a number divisible by 2? When is a number divisible by 4? divisible by 4? When is a number divisible by 5? divisible by 5? When is a number divisible by 10, When not divisible by 10, 100, &c. ?

It follows, from Prop. 5, that if any even number is divisible by 3, it is also divisible by 6. For, if an even number, it is divisible by 2; and, being divisible by 2 and by 3, it is also divisible by their product, 2X3, or 6.

PROP. VI. Every prime number, except 2 and 5, ends with 1, 3, 7, or 9: a consequence of Prop. 1 and 3.

OPERATION.

ART. 113. 1. What are the prime factors of 30? SOLUTION. If 2 is exactly contained in 30, it will be a factor of 30. By trial, it is found to be a factor. Again,

If 3 exactly divides 30, it will be a factor of it; but, since 30 is 2 times 15, if 3 is a factor of 15, it will also be a factor of 30, (Art. 111, Prin. 1.)

2)30

3)15
5

To ascertain if 3 is a factor of 30, see if it is a factor of 15. Trial shows that 3 is a factor of 15; hence, it is a factor of 30.

For the same reason, whatever number is a factor of 5, is a factor of 15 and 30; but 5 is a prime number, having no factor except itself and unity; hence, the prime factors of 30, are 1, 2, 3, and 5.

2. Find the prime factors of 42. 3. Find the prime factors of 70.

RULE

Ans. 1, 2, 3, 7.
Ans. 1, 2, 5, 7.

FOR RESOLVING A COMPOSITE NUMBER INTO PRIME FACTORS.

Divide the given number by any prime number that will exactly divide it; divide the quotient in the same manner, and so continue to divide, until a quotient is obtained which is a prime number; the last quotient and the several divisors will constitute the prime factors of the given number.

REM.-1. It will generally be most convenient to divide, first by the smallest prime number that is a factor.

REVIEW.-112. By what is every composite number divisible? Why? When any even number is divisible by 3, by what is it also divisible? With what figures do all prime numbers, except 2 and 5, terminate? 113. Find the prime factors of 30, and explain the process. What is the rule for resolving a number into prime factors?

2. The least divisor of any number is a prime number; for, if it were composite, it might be separated into factors, which would be still smaller divisors of the given numbers. Art. 111, Prin. 1.

Hence, the prime factors of any number may be found, by first dividing it by the least number that will exactly divide it; then divide the quotient as before, and so on.

3. Since 1 is a factor of every number, either prime or composite, it is not usually specified in reckoning the factors of a number.

To find the prime factors common to two numbers, resolve each into prime factors: then take the factors common to both.

REVIEW.-113. REM. 1. What prime factor should be first taken as a divisor? 2. Why is the least divisor of any number a prime number? REM. 3. Why is unity not reckoned among the prime factors of a number? How may the prime factors common to two numbers be found?