PROPERTIES OF NUMBERS. 79. SIGNS. signifies plus, or more. signifies greater than. signifies less than. () parenthesis, and RECAPITULATION. = signifies equal to. .. signifies therefore. vinculum, signify that the same op eration is to be performed upon all the quantities thus connected. DEFINITIONS. 80. Numbers are either integral or fractional. 81. Integral numbers, or Integers, are whole numbers. 82. Fractional numbers are parts of whole numbers. 83. A Factor or Divisor of a number is any number which is contained in it without a remainder; thus, 2 is a factor of 6. 84. A Prime Number is a number which contains no integral factor but itself and 1; as, 1, 2, 3, 11. 85. A Composite Number is a number which contains other integral factors besides itself and 1; as, 4, 6, 8, 25. 86. A Prime Factor is a factor which is a prime number. 87. A composite number equals the product of all its prime factors; thus, 12 = 2 × 2 × 3. 88. Two numbers are said to be prime to each other when they contain no common factor except 1; thus, 8 and 15 are prime to each other. 89. The Power of a number is the number itself, or the product obtained by taking that number a number of times as a factor. The number itself is the first power; if it is taken twice as a factor, the product is called the second power, or square; if three times, it is called the third power, or cube; if four times, the fourth power, &c. Thus, the second power of 3 is 3 X 3 =9; the third power of 3 is 3 × 3 × 3 = 27; the fifth power of 3 is 3 X 3 X 3 X 3 X 3243. 90. The Index or Exponent of a power is a figure which shows how many times the number is taken as a factor. It is written at the right of the number, and above the line. Thus, in 53, 72, 24, the exponent 2 shows that 5 is taken three times as a factor, 2 that 7 is taken twice, and 4 that 2 is taken four times as a factor. 91. The Root of a number is one of the equal factors which produce that number. If it is one of the two equal factors, it is the second, or square root; if one of the three, the third, or cube root; if one of the four, the fourth root, &c. Thus the square root of 9 is 3, the cube root of 125 is 5. 92. √ is the Radical Sign, and, by itself, denotes the square root; with a figure placed above, it denotes the root of that degree indicated by the figure; thus, signifies the third root, the sixth root. DIVISIBILITY OF NUMBERs. 93. (1.) Any number whose unit figure is 0, 2, 4, 6, or 8, is even. (2.) Any number whose unit figure is 1, 3, 5, 7, or 9, is odd. (3.) Any even number is divisible by 2. (4.) Any number is divisible by 3 when the sum of its digits is divisible by 3; thus, 2814 is divisible by 3, for 2+8+1+4 = 15, is divisible by 3. (5.) Any number is divisible by 4, when its tens and units are divisible by 4; for, as 1 hundred, and consequently any number of hundreds, is divisible by 4, the divisibility of the given number by 4 must depend upon the tens and units; thus, 86324 is divisible by 4, while 6831 is not. (6.) Any number is divisible by 5 if the units' figure is either 5 or 0; for, as 1 ten, and consequently any number of tens, is divisible by 5, the divisibility of the given number by 5 must depend upon the units. (7.) Any number is divisible by 6, if divisible by 3 and by 2. (8.) Any number is divisible by 8, if its hundreds, tens, and enits are divisible by 8; for, as 1 thousand, and consequently any number of thousands is divisible by 8, the divisibility of the given number by 8 must depend on the hundreds, tens, and units. (9.) Any number is divisible by 9 if the sum of its digits is divisible by 9*; thus, 368451 is divisible by 9, and 23476 is not. (10.) Any number is divisible by 10, 100, or 1000, if it contain at the right 1, 2, or 3 zeros; and so on. (11.) Any number is divisible by 11, if the difference between the sums of the alternate digits is 0, or a number divisible by 11; thus, in 126896, as (1+6+9) — (2+8+6) = 0, the number is divisible by 11; and in 9053, as (9+5)-(0+3)= 11, the number is divisible by 11. (12.) A number is divisible by any composite number, if it is divisible by all the factors of that number. 93. There are no rules of sufficient practical importance for determining when numbers are divisible by other numbers than those spoken of above. Their divisibility must be ascertained by trial. To do this, Divide the number successively by higher and higher primes, until one is found which divides it, or until the quotient is smaller than the divisor. If no divisor is then found, the number is prime; for, if a number contain any prime factor greater than its square root, its corresponding factor must be less. 94. If the odd numbers are written in order, and every third one from 3, every fifth one from 5, every seventh one from 7, and so on, be marked, and the figures 3, 5, 7, &c., be written Ander the figures as they are marked, the remaining numbers will be primes, and those marked will have for their factors the numbers written beneath; † thus, Eratosthenes, in the third century B. C., discovered this method of anding primes and factors of numbers, and as he made his table of parchment, cutting out the composite numbers as he found them, this parchment was called Eratosthenes' Sieve. TABLES OF PRIME AND COMPOSITE NUMBERS. 59 By applying this principle, a table can easily be made of the primes and of the composites, with their factors. TABLE OF PRIME NUMBERS TO 1201. TABLE OF THE COMPOSITE NUMBERS TO 917, Which contain no prime factor less than 7 (excepting 1*). FACTORING OF NUMBERS. 95. ILLUSTRATIVE EXAMPLE, I. Resolve 48 into its prime factors. OPERATION. 48 6 X 8; 6=2 X 3; 2 × 2 × 2 × 3, or 24 × 3. 8 = 2 X 2 X 2;... 482 X Hence, RULE I: To resolve a number into its prime factors. First separate it into any two factors; separate these factors, if they are composite, into others, and so on, till all are prime. PROOF. Multiply the factors thus obtained together, and the product, if the work is correct, will equal the given number. 96. EXAMPLES. Resolve the following numbers into their prime factors. 97. ILLUSTRATIVE EXAMPLE, II. Resolve 42075 into its prime factors. OPERATION. 3) 42075 3)14025 5) 4675 5) 935 11) 187 17 ins. 32, 52, 11, 17. Here we divide, successively, by such prime numbers as will leave no remainder, till we obtain a prime number for a quotient; since the product of these prime numbers, 3, 3, 5, 5, 11, and 17 equals the given number, they must be the prime factors of that number. Hence, RULE II. Divide th number by any prime number which is ontained in it without a remainder. Divide the quotient in the same manner, and thus continuc till a quotient is obtained which is a prime number. This quotient and the several divisors are the prime factor. |