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LAST REMAINDER, UNTIL SOME BEMAINDER IS LEFT. THE DIVISOR WHICH LEAVES NO REMAINDER IS THE GREATEST COMMON MEASURE.
If it be required to find the greatest common measure of several numbers, find first the greatest common measure of two of them, then the greatest common measure of this common measure and a third, then the greatest common measure of this and a fourth, &c. The last common measure, will be that required. Thus,
8. Find the greatest common measure of 12 and 30 and 9.
6 is the greatest common measure of 12 and 30. 3 is the greatest common measure of 6 and 9, and of course of the three given numbers 12, 30 and 9.
11. Find the greatest com. meas. of 72 and 96. A. 14. Of 330 and 462. A. 66. Of 126 and 342. A. 18. Of 84, 28 and 42. A. 14. Of 34, 746, 69,492 and 38,433. A. 3. Of 1,872, 3,456. 98,712 and 531,711. Of 12,572, 92,4:6, 2,354 and 34,564.
This principle may be applied with advantage to the reducing of Fractions to lower terms. Take the following examples.
12. Reduce to its lowest terms. 13 is the greatest com. meas.
13. Reduce 84 12 is the greatest com. meas. A. 11. 14. Reduce 15; 14; 1 ; and 481.
The pupil has, no doubt, by this time observed that there are some numbers which have no co mon measure greater than one. These numbers are said to be PRIME TO EACH OTHER.
This definition must not be confounded with that given of prime numbers ; for num. bers which are themselves composite, may be prime to each other. Thus 25 and 27, both of which are composite, are prime to each other. A number is prime to itself, which has no measure greater than 1. Numbers are prime to each other, which have no common measure greater than 1.
We may here make one more suggestion, which will be useful in finding the measures of numbers by trial. 2 and 5, which are prime to each other, are measures of 20. If 20 be resolved into the factors, 5 and 4, we know that 2, which is a measure of 20, must be a measure of 4, since it is not a measure of 5, and has no factor, which is a measure of 5." · 20 may therefore be divided successively by 5 and 2, and, consequently, ($ XXIX.) by their product 10. The same may be shown of any two or more numbers, prime to each other, which are measures of a third number. . Hence, If a number be divisible by two or more numbers, which are prime to each other, it is likewise divisible by their product. If the numbers are not prime to each other, we cannot make this inference.
MENTAL EXERCISES. Ş XLIV. 1. Change the Fractions and to others having the same denominator. This can easily be done, because we know that } is the same as 4.
2. Reduce the Fractions į and to the same denominator.
3. Reduce and to the same denominator..
By a COMMON DENOMINATOR is meant the same denominator. 4. Reduce 4 and iš to a common denominator. į and
and į : and it's and it : * and When the numbers are larger the process cannot be performed in the mind. For example.
5. Reduce a's and all to a common denominator. When we multiply both terms of a Fraction by the same number, the value is not altered. ($ xli.) Then we may multiply the 3, and the 24, in the last Fraction by 23, without altering the value. And, we may multiply the 1 and the 23 of the first Fraction by 24, without altering the value.
Thus, 23X3=69; and 23X24=552.
Therefore, a=aAnd 24X1=24; and 24 X 23= 552. Therefore, 2= But, on and have the same denominator.
In this process, we multiplied both terms of each Fraction, by the denominator of the other Fraction.
6. Reduce in and 74 to a com. denom. 11 and if;
and 11; 44 and it ; 5 and 1. It will be found most convenient, to reduce the Fractions first, to their lowest terms. 7. Reduce 4 and I to a common denominator.
4=;, and fi=1. A. and . 8. Reduce 1st and to a com. den. Hi; and :
and 74. 9. Reduce i and band } to a com. den.
A., and Here we have three Fractions. If there were only the first two, viz. f and s, we should multiply the terms of by 6 and those of 5 by 4. But if we should multiply these terms, thus increased, by the other denominator, 2, it would not alter the value of the Frac. tions. And, if, then, we should multiply the terms of by 4 and 6, it would not alter its value. Therefore to reduce Fractions to a common denominator,
MULTIPLY BOTH TERMS OF EACH FRACTION BY THE DENOMINATORS OG ALL THE OTHER FRACTIONS.
NOTE. As all the denominators of the reduced Fractions are alike, when the denominator of one has been found it may be written down for all the others, without the trouble of Multiplication.
EXAMPLES FOR PRACTICE. 10. Reduce it, ib, and 17 to a com. den. ☺and *; 3, , i and ; , , }, and ; de , and 1 ; th, i'q, is, T and ..
8 XLV. A much smaller common denominator, may often be found, than the preceding rule would give. To determine what the smallest possible denominator would be, we must attend to the subject of multiples. 15 contains 3, an even number of times exactly. Therefore, 15 is called a multiple of 3.
A NUMBER IS CALLED A MULTIPLE OF ANY NUMBER WAICH WILL DIVIDE IT WITHOUT A REMAINDER.
It will be seen that multiple and measure are merely relative terms. Any number is the multiple of its measure, and the measure of its multiple. 15 is a multiple of both 3 and 5. For 3 and 5 are measures of 15. Hence 15 is called a common multiple of 3 and 5.
ANY MULTIPLE OF TWO OR MORE NUMBERS, IS CALLED THEIR COMMON MULTIPLE.
If any two or more numbers be multiplied togther, the product is evidently a common multiple of all; for it may be divided by either of the factors which compose it. If two numbers be prime to each other, their product is likewise their least common multiple. Thus 3 and 5 are prime to each other, and their product, 15, is their least common multiple. For if you suppose that 3, repeated fewer times, will contain 5, you must suppose that 3 contains some factor, which, multiplied by 4 will produce 5. But, as 3 and 5 are prime to cach other, they have no common factor. Therefore 5 times 3=15, is the least common multiple of 3 and 5. If there be more num. bers than two, the same is true for similar reasons.
Take a case, in which the numbers are not prime to each other. 15 and 10 have 5 for their greatest common measure.
Divide one of them, as 15, by this common measure, and the quotient is 3, Multiply this into the other, and the product, 30, is the least common multiple of 15 and 10. For if you suppose that 10, repeated fewer times, as, for example 2 times, will contain 15, you must suppose that 10 contains some factor, which multiplied by 2, will produce 15.
But this cannot be, for the quotient 3 was found by dividing 15 by the greatest common measure of the two numbers. Therefore, 3 times 10=30, is the least common multiple of 15 and 10.
Hence, to find the least common multiple of two numbers,
DIVIDE ONE OF THE NUMBERS BY THEIR GREATEST COMMON MEASURE, AND MULTIPLY THE QUOTIENT BY THE OTHER.
EXAMPLES FOR PRACTICE. 1. Find the least common multiple of 12 and 16. Their greatest com. meas. is 4. 16-434. 4x12=48 Ans.
2. Find the least common multiple of 14 and 18. A. 126. Of 9 and 12. A. 36. Of 24 and 30° A. 120. Of 25 and 45. Of 24 and 84. Of 36 and 48. Of 26 and 36, Of 32 and 56. Of 81 and 108.
When there are more than two numbers, after finding the least common multi. ple of two of them, proceed with this multiple and the third number, to find another common multiple by the rule. After finding for three, proceed in the same manner with the fourth number, and so on.
3. Find the least common multiple of 15, 18 and 24. The least com. mult. of 15 and 18 is 90; of 90 and 24, 540 Ans.
4. Find the least com. mult. of 12, 16 and 30. A. 240. Of 9, 12, 16, 18 and 24. A. 144. Of 6, 10, 25 and 36. Of 224, 648, 936 and 872. Of 828, 333, 756 and 963.
When there are many numbers and particularly if they are large, it is tedious to proceed as above. There is a shorter process.
We have already seen, that when several numbers are prime to each other, their product is their least common multiple. ff, then, we can contrive to find all the factors prime to each other, in the several numbers given, the product of these factors will be the least common multiple, not only of the factors, themselves, but also of the numbers from which we obtained them. These factors must be found by actual trial, thus. Place the numbers in a row and divide as many of them as possible by any numbor, which will divide them without a remainder, placing the quotients, with the numbers not divided, likewise in a row under the first. Divide as many numbers in this row as possible, in the same manner as before, and continue this process until no two numbers can be divided by any number greater than 1. The divisors used, and the last row of numbers, will be the factors sought. Of course,
THE PRODUCT OF THESE FACTORS WILL BE THE LEAST COMMON MULTIPLE REQUIRED.
5. For example: find the least common multiple of 72, 64, 21, 18, and 98. I observe that all, but 21, may
2 | 72 ; 64 ; 21 ; 18 ; 98 be divided by 2. Therefore, I 3 36 ; 32 ; 21; 9; 49 divide by 2 and bring dɔwn 21. 3 | 12 : 32 ;
7; 3; 49 Then, that all the next row, ex.
71 4; 32 ; 7 ; 1; 49 cept 32 and 49, may be divided by 3. Therefore I divide by 3, and
4 | 4; 32 1; 7 bring down 32 and 49. In this 1; 8 ; 1; 1.; 7 way, I proceed, dividing, each time as many as possible. I obtain then the factors 2, 3, 3, 7,4, 1, 8, 1, 1, 7. Therefore 2X3X3X7X 4x8X7=28,224 is the least common multiple of the given numbers. In multiplying, I neglect the factors, 1, 1, 1, since they will not al. ter the product. The intelligent pupil will at once reply that these factors are not all prime to each other. But, it should be recollected, that, by this process, the factors of some of the numbers become resolved into their component parts. If these component parts be re-multiplied, the compound factors, thus obtained, will be all prime to each other. Thus, the two factors, 3 and 3, were derived, suc. cessively from 18. Therefore the compound factor of 18, is 3X3= 9. In like manner 2, 4 and 8 were derived successively from 64, and the compound factor is 2X4X8=64. So 7 and 7 were derived from 98, and the compound factor is 49. It will now be seen, that these compound factors, 64, 9 and 49, are all prime to each other; and that they contain all the factors, which are component parts of the given numbers, and no more,
By- the process, it will be seen, that when any number is a factor of several numbers, it is excluded from them all, by Division, and is then made once a firctor in the multiple obtained. If all the given numbers of which it is a factor be not divided by it, then we shall employ it more than once as a factor in the result, and thus make the multiple obtained too large. From this consideration, the pupil will see the necessity of the following caution in employing the atove mode of operation.
In making each division, choose divisors which will divide AS MANY
NUMBERS AS POSSIBLE.
This will be best understood by illustration. 6. Find the least com. mult. of 6, 9, and 24.
6 and 24 can be divided by 9. But all the numbers can be divided by 3. If, then we first divide by 6, we shall leave the 9 undivided by 3, and so obtain too great a multiplier. First divide by 6.
First divide by 3. 66, 9, 24,
316, 9, 24, | 1,. 9,
22, 3, 8, We can divide no further. Therefore we should expect 6x9
| 1; 3,
4, X4=216 to be the least com. Hence, 3X2X3X4=72 is the mult., which the other operation least com. mult. shows to be incorrect.
None of our common Arithmetics provide against this source of error, and hence the learner is often perplexed by finding his result wrong, when he has strictly followed the directions of his rule. By this mode, perform the following.
7. Find the least com. mult. of 32, 72 and 120. A. 1,440. Of 30,48 and 56. A. 1,680. Of 250, 180 and 540. A. 13,500. Of 375, 125, 320 and 45. Of 872, 16, 98 and 75. Of 196, 762, 1,131, 340 and 460.
In finding a common denominator for several Fractions, we mul. tiply all the denominators together. The common denominator, therefore, is a common multiple of the given denominators. Or course, the least com. den. must be the least com. mult.
8. Reduce and to the least com. den.
24 is the least com. mult. of 5 and 8. We must therefore bring the Fractions to 24ths. 24 24ths make a whole one. Then of 1 is of 24 24ths=1; and of 1 is of 24 24ths= A. it and 24:
Hence, for the numerators of the reduced Fractions, multiply the com. denom. by each of the given Fractions. ($ xxx11.)
9. Reduce and to the least com. den. A. and stand A. and 1. Dia and A. 4524 and , tap and 3 6 15,1 and 1 os, s, ils and als: