2. What is the greatest common measure of 612 and Ans. 36. 540? 3. What is the greatest common measure of 720, 336 and 1736 ? Ans. 8. PROBLEM II. 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. 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 ex ample, thus: it is evident, that 3×5×8×10=1200 may be divided by 3, 5, 8, and 10, without a remainder; but 10 is a multiple of 5, therefore 3×5×8×2, or 240, is also divisible by 3, 5, 8, and 10. Also 8 is a multiple of 2; therefore 3×5×4× 2=120 is also divisible by 3, 5, 8, and 10; and is evidently the least number that can be so divided. 1 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 Ans. 2520. pine digits, without a remainder ? 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 divisi ble by 3, or 9, the whole is divisible by 3, or 9. 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. EXAMPLES. 1. Reduce to its lowest terms. (2) (3) (2) (2) 240 (2) 744240 72 36 the answer, Or thus : 144)240(1 96)144( 48)96(2 Therefore 48 is the greatest common measure, and 48), 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 numbers must be divided. Thus 4+8+10=2+4+5=11. 2 9. But if the numbers have the sign of multiplication between them, only one of them must be divided. Thus 3×8×10 3X4X10_IX4X10_IX2X10_20=20. 2x6 To reduce a mixed number to its equivalent improper fraction. RULE.* Multiply the whole number by the denominator of the fraction, and add the numerator to the product, then that sum written above the denominator will form the fraction tequired. * All fractions represent a division of the numerator by the de. nominator, and are taken altogether as proper and adequate expressions for the quotient. Thus the quotient of 2 divided by 3 is; from whence the rule is manifest; for if any number is multiplied and divided by the same number, it is evident the quotient must be the same as the quantity first given. 2. Reduce 183 to its equivalent improper fraction. 3. Reduce 5147 to an improper fraction. 4. Reduce 100% to an improper fraction. 5. Reduce 47to an improper fraction. CASE III. To reduce an improper fraction to its equivalent whole or mix ed number. RULE.* Divide the numerator by the dennominator, and the quotient will be the whole or mixed number required. EXAMPLES. 1. Reduce to its equivalent whole or mixed number. 16 16)981(61 21 16 5 Or, 56 1245 22 981÷16=61 the answer. 2. Reduce to its equivalent whole or mixed number. Ans. 7. 3. Reduce to its equivalent whole or mixed numAns. 56. 4. Reduce 3848 to its equivalent whole or mixed numAns. 183 to its equivalent whole or mixed ber. ber. 5. Reduce number. 621611 514 2087 Ans. 1209 CASE * This rule is plainly the reverse of the former, and has its reason in the nature of common division. |