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 um 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 S.

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. 11. How may the smaller composite numbers be resolved into prime factors ? What are the prime factors of C? Of 8? Of 9? Of 10? 111. In determining the factors of a number, what two principles are ned? Explain the årst 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.

12. 24. Ans. 2, 2, 2, 3. | 21. 98.

22. What are the prime factors of 105?

23. Of 168?

Ans. 2, 7, 7.

Ans. 3, 5, 7.

Ans. 2, 2, 2, 3, 7.

Ans. 2, 2, 2, 3, 3, 3.

Ans. 2, 3, 5, 11.

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? flow may the prime factors common to two numbers be found?

ART. 114. Since any composite number is divisible not only by each of its prime factors, but also by the product of any two or more of them, (Art. 112, Prop. V.),

Therefore, to find the several divisors of a composite number, resolve it into its prime factors, and form from them as many different products as possible.

Thus, 302 X3 X5, and its several divisors are 2, 3, 5, and 2X3, 2X5, and 3X5.

IX. GREATEST COMMON DIVISOR.

ART. 115. A divisor or measure of a number (Art. 110, Def. 8), is a number that will divide it without a remainder; thus, 2 is a divisor of 4; 3 of 6, &c.

A common divisor of two or more numbers, is a number that will divide each without a remainder; thus, 2 is a common divisor of 12 and 18.

The greatest common divisor of two or more numbers, is the greatest number that will divide each without a remainder; thus, 6 is the greatest common divisor of 12 and 18.

REM.-Two numbers may have several common divisors, but only one greatest common divisor.

G. C. D. should be read, greatest common divisor.

REVIEW.-114. How may the several divisors of a composite number be found? Why? 115. What is a divisor of a number? Give examples. 115. What is a common divisor of two or more numbers? Give example. What the greatest common divisor? Give example.