8. A prime number is that, which can only be measured by itself, or an unit. 9. That number, which is produced by multiplying several numbers together is called a composite number. 10. A perfect number is equal to the sum of all its alir quot parts. PROBLEM 1. 1 To find the greatest common measure of twa, or more numbers. RUL E. 1. If there be two numbers only, divide the greater by the less, and this divisor by the remainder, and so on, always dividing the last divifor by the last remainder, till nothing remain, then will the last divisor be the greatest common measure required. 2. When there are more than two numbers, find the greatest common measure of two of them, as before 3 then of that common measure and one of the other numbers, and so on, through all the numbers, to the last then will the greatest common measure, last found, be the answer. 3. If i happens to be the common measure, the given numbers are prime to each other, and found to be incommenfurable, or in their lowest terms. EXAMPLES EXAMP LTE S. 1. What is the greatest common measure of 1836, 3996, and 1044 ? So 108 is the greatest 1836)3996(2 common measure of 3672 3996 and 1836 Hence 108) 10449 324)183615 972 1620 72 ) 10811 216) 324(1 72 216 Lait gr. com. meal. 36)72(2 Com. meaf. 108)216(2 72 216 Therefore, 36 is the answer required. 2. What is the greatest common measure of 1224 and 1080 ? Anf: 72. PROBLEM II. To find the least common multiple of two, or more numbers, RUL E. 1. Divide by any number that will clivide two or more, of the given numbers, without a remainder, and fut the quotients, together with the undivided numbers, in a line beneath. 2. Divide the second line, as before, and so on, till there are no two numbers, that can be divided ; then, the continued product of the divisors and quotients, will give the multiple required. EXAMPLES. E X A M P L E s. 20 2 I I * I 1. What is the least common multiple of 6, 10, 16 and 20? 1 survey my given *5)6 10 16 numbers and find that five will divide two of *2)6 16 4 them, viz. 10 and 20, which I divide by 5, *2) 3 8 2 bringing, into a line with the quotients, the *3 *4 numbers, which 5 will not measure : Again, 1 view the numbers in 5 X 2 X 2 X 3 X 4 = 240 Ans. the second line, and find 2 will measure them all, and I get 3, 1, 8, 2, in the third line, and find that 2 will measure 8 and 2, and in the fourth line, get 3, 1, 4, 1, all prime, I then multiply the prime numbers and the divisors continually into each other, for the number fought, and find it to be 240. 2. What is the least common multiple of 6 and 8 ? Anf. 24. * * * 3. What is the least number that 3, 5, 8 and 10 will measure ? Anf. 120. 4. What is the least number which can be divided by the 9 digits, separately, without a remainder ? Ans. 2520 REDUCTION of VULGAR FRACTIONS Is the bringing of them out of one form into another, in order to prepare them for the operations of Addition, Subtracion, &c. CASE To abbreviate, or reduce fractions to their lowest ternis. Ru L E. Divide the terms of the given fraction by any num. ber, which will divide them without a remainder, and the quotients, again, in the same manner ; and so on, till it appears that there is no number greater than 1, which * That dividing both the terms, that is, both numerator and denominator of the fraction, equally by any number whatever, will give another fraction, equal to the former, is evident : And if those divisions be performed as often as can be done, or the common divisor be the greatest possible, the terms of the resulting fraction must be the least poslible. 2. Note I. Any number, ending with an even number of cypher, is divisible by 2 Any number, ending with 5 or o, is divisible by 5. 3. If the right hand place of any number be o, the whole is die visible by 10. If the two right hand figures of any number be divisible by 4, the whole is divisible by 4. 5. If the three right hand figurcs of any number be divisible by 8, the whole is divisible by 8. 6. If the sum of the digits, conftituting any number, be divisible by z or 9, the whole is divisible by 3 or 9. 1. If a number cannot be divided by some number less than the square root thereof, that number is a prime. 8. All prime numbers, except 2 and 5, have 1, 3, 7, or g in the place of units; and all other numbers are cimp site. 9. When numbers, with the fign' of Addition or Subtraction be tween them, are to be divided by any number, each of the numbers must be divided : Thus, 6+9+12=2+3+4=9. 3 But if the numbers have the sign of Multiplication between them ; then only one of them must be divided : Thus, 2 X 24 24. 2 X 5 IX5 IX 5 which will divide them, and the fraction will be in its lowest terms. Or, Divide both the terms of the fraction by their greateit common measure, and the quotients will be the terms of the fraction required. EX A M P L E S. 8 Reduce 338 to its lowest termo. (+) (3) 1s = . Or thus' : 288)480(1 Therefore 96 is the greatest common 288 measure. 192)288(1 288= the same as before. 192 and 96 oo} Com. meal. 90)192(2 192 57 2. Reduce to its lowest terms. 456 3. Reduce to its lowest terms. 181 4. Reduce to its lowest.terms. 2858 46 Anf. s. Anf. And: I 4 29 To reduce a mixed number to its equivalent improper Fraction, RU * L E. Multiply the whole number by the denominator of the fraction, and add the numerator of the fraction to the product ; under which fubjoin the denominator, and it will form the fraction required. EXAMPLES, 51, * All fractions represent a divifion of a numerator by the denominator, and are taken altogethex as proper and adequate expressions of the quotient. Thus the quotient of 3, divided by 4, is : |