Al the numbers that are divisible by 4; and from the manner of forming this series, it is evident, 1st. That the product of any one number of the series by any integral number whatever, will contain 4 an exact number of times; 2d. The sum of any two numbers of the series will contain 4 an exact number of times; and 3d. The difference of any two will contain 4 an exact number of times. Hence, I Any number which will exactly divide one of two numbers will divide their product. II. Any number which will exactly divide each of two numbers will divide their sum. III. Any number which will exactly divide each of two numbers will divide their difference. 136. From these principles we derive the following properties: I. Any number terminating with 0, 00, 000, etc., is divisible by 10, 100, 1000, etc., or by any factor of 10, 100, or 1000. For by cutting off the cipher or ciphers, the number will be divided by 10. 100. or 1000, etc., without a remainder, (122); and a number of which 10, 100, or 1000, etc., is a factor, will contain any factor of 10, 100, or 1000, etc., (I). II. A number is divisible by 2 if its right hand figure is even or divisible by 2. For, the part at the left of the units' place, taken alone, with its local value, is a number which terminates with a cipher, and is divisible by 2, because 2 is a factor of 10, (I); and if both parts, taken separately, with their local values, are divisible by 2, their sum, which is the entire number, is divisible by 2, (135, II). NOTE. Hence, all numbers terminating with 0, 2, 4, 6, or 8, are even, and all numbers terminating with 1, 3, 5, 7, or 9, are odd. III. A number is divisible by 4 if the number expressed by its two right hand figures is divisible by 4. For, the part at the left of the tens' place, taken alone, with its local value, is a number which terminates with two ciphers, and is divisible by 4, because 4 is a factor of 100, (I); and if both parts, taken separately, with their local values, are divisible by 4, their sum which is the entire number, is divisible by 4, (135, II) IV. A number is divisible by 8 if the number expressed by its three right hand figures is divisible by 8. For, the part at the left of the hundreds' place, taken alone, with its local value, is a number which terminates with three ciphers, and is divisible by 8, because 8 is a factor of 1000, (I); and if both parts, taken separately, with their local values, are divisible by 8, their sum, or the entire number, s divisible by 8, (135, II). V. A number is divisible by any power of 2, if as many right hand figures of the number as are equal to the index of the given power, are divisible by the given power. For, as 2 is a factor of 10, any power of 2 is a factor of the corresponding power of 10, or of a unit of an order one higher than is indicated by the index of the given power of 2; and if both parts of a number, taken separately, with their local values, are divisible by a power of 2, their sum, or the entire number, is divisible by the same power of 2, (135, II). VI. A number is divisible by 5 if its right hand figure is 0, or 5. For, if a number terminates with a cipher, it is divisible by 5, because 5 is a factor of 10, (I); and if it terminates with 5, both parts, the units and the figures at the left of units, taken separately, with their local values, are divisible by 5, and consequently their sum, or the entire number, is divisible by 5, (135, II). VII. A number is divisible by 25 if the number expressed by its two right hand figures is divisible by 25. For, the part at the left of the tens' figure, taken with its local value, is a number terminating with two ciphers, and is divisible by 25, because 25 is a factor of 100, (I); and if both parts, taken separately, with their local values, are divisible by 25, their sum, or the entire number, is divisible by 25, (135, II). VIII. A number is divisible by any power of 5, if as many right hand figures of the number as are equal to the index of the given power are divisible by the given power. For, as 5 is a factor of 10, any power of 5 is a factor of the corresponding power of 10, or of a unit of an order one higher than is indi ted by the index of the given power of 5; and if both parts of a number, taken separately, with their local values, are divisible by a power of 5, their sum, or the entire number, is divisible by the same power of 5, (135, II). IX. A number is divisible by 9 if the sum of its digits is divisible by 9. For, if any number, as 7245, be separated into its parts, 7000 + 200+40 +5, and each part be divided by 9, the several remainders will be the digits 7, 2, 4, and 5, respectively; hence, if the sum of these digits, or remainders, be 9 or an exact number of 9's, the entire number must contain an exact number of 9's, and will therefore be divisible by 9. NOTE. Whence it follows that if a number be divided by 9, the remainder will be the same as the excess of 9's in the sum of the digits of the number. Upon this property depends one of the methods of proving the operations in the four Fundamental Rules. X. A number is divisible by a composite number, when it is divisible, successively, by all the component factors of the composite number. For, dividing any number successively by several factors, is the same as dividing by the product of these factors, (119, I). XI. An odd number is not divisible by an even number. For, the product of any even number by any odd number is even, and, consequently, any composite odd number can contain only odd factors. XII. An even number that is divisible by an odd number, is also divisible by twice that odd number. For, if any even number be divided by an odd number, the quotient must be even, and divisible by 2; hence, the given even number, being divisible successively by the odd number and 2, will be divisible by their product, or twice the odd number, (119, I). PRIME NUMBERS. 137. A Prime Number is one that can not be resolved or separated into two or more integral factors. NOTE. Every number must be either prime or composite. 138. To find all the prime numbers within any given limit, we observe that all even numbers except 2 are composite; hence, the prime numbers must be sought among the odd numbers 139, If the odd numbers be written in their order, thus; 1, 3, 5, 7, 9, 11, 13, 15 17, etc., we observe, 1st. Taking every third number after 3, we have 3 times 3, 5 times 3, 7 times 3, and so on; which are the only odd numbers divisible by 3. 2d. Taking every fifth number after 5, we have 3 times 5, 5 times 5,7 times 5, and so on; which are the only odd numbers divisible by 5. And the same will be true of every other number in the series. Hence, 3d. If we cancel every third number, counting from 3, no number divisible by 3 will be left; and since 3 times 5 will be canceled, 5 times 5, or 25, will be the least composite number left in the series. Hence, 4th. If we cancel every fifth number, counting from 25, no number divisible by 5 will be left; and since 3 times 7, and 5 times 7, will be canceled, 7 times 7, or 49, will be the least composite number left in the series. And thus with all the prime numbers. Hence, 140. To find all the prime numbers within any given limit, we have the following RULE. I. Write all the odd numbers in their natural order. II. Cancel, or cross out, 3 times 3, or 9, and every third number after it; 5 times 5, or 25, and every fifth number after it; 7 times 7, or 49, and every seventh number after it; and so on, beginning with the second power of each prime number in succession, till the given limit is reached. The numbers remaining, together with the number 2, will be the prime numbers required. NOTES.-1. It is unnecessary to count for every ninth number after 9 times 9, for being divisible by 3, they will be found already canceled; the same may be said of any other canceled, or composite number. 2. This method of obtaining a list of the prime numbers was employed by Eratosthenes (born B. C., 275), and is called Eratosthenes' Sieve. 141. To resolve any composite number into its prime factors. The Prime Factors of a number are those prime numbers which multiplied together will produce the given number. 142. The process of factoring numbers depends upon the following principles: I. Every prime factor of a number is an exact divisor of that number. II. The only exact divisors of a number are its prime factors, or some combination of its prime factors. 1. What are the prime factors of 798? OPERATION. 2798 3 399 7 133 19 19 ANALYSIS. Since the given number is even, we divide by 2, and obtain an odd number, 399, for a quotient. We then divide by the prime numbers 3, 7, and 19, successively, and the last quotient is 1. The divisors, 2, 3, 7, and 19, are the prime factors required, (II). Hence, the |