60 least number, which can be so measured, it is called their least common multiple; thus 30, 45, 60 and 75, are multiples of 3 and 5; but their least common multiple is 15.* PROBLEM I. To find the greatest common measure of two or more numbers. RULE.† 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 divisor by the last remainder, till nothing * A prime number is that, which can only be measured by an unit. That number, which is produced by multiplying several numbers together, is called a composite number. A perfect number is equal to the sum of all its aliquot parts. The following perfect numbers are taken from the Petersburgh acts, and are all that are known at present. 6 28 496 8128 33550336 8589869056 137438691328 2305843008139952128 2417851639228158837784576 9903520314282971830448816128 There are several other numbers, which have received different denominations, but they are principally of use in Algebra, and the higher parts of mathematics. This and the following problem will be found very useful in the doctrine of fractions, and several other parts of Arithmetic. The truth of the rule may be shewn from the first example.For since 54 measures 108, it also measures 108+54, or 162. nothing remains, 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; and 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 happen to be the common measure, the given numbers are prime to each other, and found to be incommensurable. EXAMPLES. 1. Required the greatest common measure of 918, 1998 Again, since 54 measures 108, and 162, it also measures 5X162+108, or 918. In the same manner it will be found to measure 2X918+162, or 1998, and so on. Therefore 54 measures both 918 and 1998. It is also the greatest common measure; for suppose there be a greater, then since the greater measures 918 and 1998, it also measures the remainder 162; and since it measures 162 and 918, it also measures the remainder 108; in the same manner it will be found to measure the remainder 54; that is, the greater meas ures 2. What is the greatest common measure of 611 and 540 ? Ans. 36. 3. What is the greatest common measure of 720, 336 and 1736 ? PROBLEM II. Ans. 8. To find the least common multiple of two or more numbers. RULE.* 1. Divide by any number, that will divide two or more of the given numbers without a remainder, and set 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. 3. What is the least common multiple of 3, 5, 8 and 10? 5)3 5 8 10 2)3 I 8 3 I 4 5X2X3X4=120 the answer. 2. What ures the less, which is absurd. Therefore 54 is the greatest com mon measure. In the very same manner, the demonstration may be applied to 3 or more numbers. : *The reason of this rule may also be shewn from the first example, thus it is evident, that 3X5X8X10=1200 may be divided by 3, 5, 8, and 10, without a remainder; but 10 is a multiple of 5, therefore 3X5X8X2, or 240, is also divisible by 3, 5, 8, and 10. Also 8 is a multiple of 2; therefore 3X5X4X 2120 is also divisible by 3, 5, 8, and 10; and is evidently the least number that can be so divided. 2. What is the least common multiple of 4 and 6? Ans. 12. 3. What is the least number, that 3, 4, 8 and 12 will measure? Ans. 24. 4. What is the least number that can be divided by the pine digits, without a remainder? Ans. 2520. REDUCTION of VULGAR FRACTIONS. Reduction of Vulgar Fractions is the bringing them out of one form into another, in order to prepare them for the operations of addition, subtraction, &c. CASE I. To abbreviate or reduce fractions to their lowest terms. RULE.* Divide the terms of the given fraction by any number that will divide them without a remainder, and these quo tients *That dividing both the terms of the fraction equally, by any number whatever, will give another fraction equal to the former, is evident. And if those divisions are 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 possible. NOTE 1. Any number ending with an even number, or a cypher, is divisible by 2. 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 divisible by 10. 4. If the two right-hand figures of any number are divisible by 4, the whole is divisible by 4. 5. If the three right-hand figures of any number are divisible by 8, the whole is divisible by 8. 6. If the sum of the digits constituting any number be divisible by 3, or 9, the whole is divisible by 3, or 9. 7. If tients again in the same manner; and so on, till it appears that there is no number greater than 1, which will divide them, and the fraction will be in its lowest terms. Or, Divide both the terms of the fraction by their greatest common measure, and the quotients will be the terms of the fraction required. Therefore 48 is the greatest common measure, and 48)144, the same as before. 240 2. Reduce 7. All prime numbers, except 2 and 5, have 1, 3, 7, or 9, in the place of units; and all other numbers are composite. 8. When numbers, with the sign of addition or subtraction between them, are to be divided by any number, each of the numThus 4+8+10=2+4+5=11. bers must be divided. 2 9. But if the numbers have the sign of multiplication between them, only one of them must be divided. Thus 3X8X10 2X6 |