« ΠροηγούμενηΣυνέχεια »
In the last case I find the share of 3 men, and then of I In dividing by 6 there is a remainder 345, which is 345, this divided by 6 gives a fraction 345. In dividing by 3 there is a remainder 2345, which is equal to 1545, this di vided by 3 gives a fraction 15, and the answer is $411546 each.
From these examples we derive the following rule: When the divisor is a compound number, separate the divisor into two or more factors, and divide the dividend by one factor of the divisor, and that quotient by another, and so on, until you have divided by the whole, and the last quotient will be the quotient required.
When there are zeros at the right of the divisor, you may cut them off, and as many figures from the right of the dividend, making the figures so cut off the numerator of a fraction, and 1 with the zeros cut off, will be the denominator; then divide by the remaining figures of the divisor.
XXI. In Art. XIX, it was observed, that if both the numerator and denominator of a fraction can be divided by the same number, without a remainder, it may be done, and the value of the fraction will remain the same. This gives rise to a question, how to find the divisors of numbers.
It is evident that if one number contain another a certain number of times, twice that number will contain the other twice as many times; three times that number will contain the other thrice as many times, &c. that if one number is divisible by another, that number taken any number of times will be divisible by it also.
10 (and consequently any number of tens) is divisible by 2, 5, and 10; therefore if the right hand figure of any number is zero, the number may be divided by either 2, 5, or 10. If the right hand figure is divisible by 2, the number may be divided by 2. If the right hand figure is 5, the number may be divided by 5.
100 (and consequently any number of hundreds) is divisi
ble by 4; therefore if the two right hand figures taken together are divisible by 4, the number may be divided by 4.
200 is divisible by 8; therefore if the hundreds are even, and the two right hand figures are divisible by 8, the number may be divided by 8. But if the hundreds are odd, it will be necessary to try the three right hand figures. 1000, being even hundreds, is divisible by 8.
To find if a number is divisible by 3 or 9, add together all the figures of the number, as if they were units, and if the sum is divisible by 3 or 9, the number may be divided by 3 or 9.
The number 387 is divisible by 3 or 9, because 3+8 +7=18, which is divisible by both 3 or 9.
The proof of the above rule is as follows: 10=9+1; 20=2×9+2; 30=3 × 9+3; 525x9+5+ 2; 10099+1; 200 = 2 × 99 + 2; 387 = 3 × 99+ 3+8×9 +8 + 7 = 3 × 99 + 8 × 9 + 3 + 8 + 7. That is, in all cases, if a number of tens be divided by 9, the remainder will be equal to the number of tens; and if a number of hundreds be divided by 9, the remainder will always be equal to the number of hundreds. The same is true of thousands and higher numbers. Therefore, if the tens, hundreds, thousands, &c. of any number be divided separately by 9, the remainders will be the figures of that number, as in the above example 387. Now if the sum of these remainders be divisible by 9, the whole number must be so. But as far as the number may be divided by 9, it may be divided by 3; therefore, if the sum of the remainders, after dividing by 9, that is, the sum of the figures are divisible by 3, the whole number will be divisible by 3.
The numbers 615, 156, 3846, 28572 are divisible by 3, because the sum of the figures in the first is 12, in the second 12, in the third 21, and in the fourth 24.
The numbers 216, 378, 6453, and 804672 are divisible by 9, because the sum of the figures in the first is 9, in the second 18, in the third 18, and in the fourth 27.
When a number is divisible by both 2 and 3, it is divisible by their product 6. If it is divisible by 4 and 3 or 5 and 3, it is divisible by their products 12 and 15. In fine, when a number is divisible by any two or more numbers, it is divisible by their product.
N. B. To know if a number is divisible by 7, 11, 23, &c. it must be found by trial,
When two or more numbers can be divided by the same number without a remainder, that number is called their common divisor, and the greatest number which will divide them so, is called their greatest common divisor. When two or more numbers have several common divisors, it is evident that the greatest common divisor will be the product of them all.
In order to reduce a fraction to the lowest terms possible, it is necessary to divide the numerator and denominator by all their common divisors, or by their greatest common divisor at first.
Reduce 1 to its lowest terms.
I observe in the first place that both numerator and denominator are divisible by 9, because the sum of the figures in each is 9. I observe also, that both are divisible by 2, because the right hand figure of each is so; therefore they are both divisible by 18. But it is most convenient to divide by them separately.
7 and 19 have no common divisor, therefore cannot be redued to lower terms.
The greatest common divisor cannot always be found by the above method. It will therefore be useful to find a rule by which it may always be discovered.
Let us take the same numbers 126 and 342.
126 is a number of even 18s, and 342 is a number of even 18s; therefore if 126 be subtracted from 342, the remainder 216 must be a number of even 18s. And if 126 be subtracted from 216, the remainder 90 must be a number of even 18s. Now I cannot subtract 126 from 90, but since 90 is a number of even 18s, if I subtract it from 126, the remainder 36 must be a number of even 18s. Now if 36 be subtracted from 90, the remainder 54 must be a number of even 18s. Subtracting 36 from 54, the remainder is 18. Thus by subtracting one number from the other, a smaller number was obtained every time, which was always a number of even 18s, until at last I came to 18 itself. If 18 be subtracted twice from 36 there will be no remainder. It is easy to see, that whatever be the common divisor, since each number is a certain number of times the common divisor, if one be subtracted from the other, the remainder will be a certain number of times the common divisor, that is, it will have the same divisor as the numbers themselves. And every
time the subtraction is made, a new number, smaller than the last, is obtained, which has the same divisor; and at length the remainder must be the common divisor itself; and if this be subtracted from the last smaller number as many times as it can be, there will be no remainder. By this it may be known when the common divisor is found. It is the number which being subtracted leaves no remainder. When one number is considerably larger than the other, division may be substituted for subtraction. The remainders only are to be noticed, no regard is to be paid to the quotient.
Reduce the fraction 239 to its lowest terms.
Subtracting 330 from 462, there remains 132. 132 may be subtracted twice, or which is the same thing, is contained twice in 330, and there is 66 remainder. 66 may be subtracted twice from 132, or it is contained twice in 132, and leaves no remainder; 66 therefore is the greatest common divisor. Dividing both numerator and denominator by 66, the fraction is reduced to 4.
From the above examples is derived the following general rule, to find the greatest common divisor of two numbers: Divide the greater by the less, and if there is no remainder, that number is itself the divisor required; but if there is a remainder, divide the divisor by the remainder, and then divide the last divisor by that remainder, and so on, until there is no remainder, and the last divisor is the divisor required.
If there be more than two numbers of which the greatest common divisor is to be found; find the greatest common divisor of two of them, and then take that common divisor and one of the other numbers, and find their greatest common divisor, and so on.
Reduce the fraction to its lowest terms.
17 (9 9
1 is the greatest common divisor in this example.
Therefore the fraction
cannot be reduced.
XXII. The method for finding the common denominator, given in Art. XIX. though always certain, is not always the best; for it frequently happens that they may be reduced to a common denominator, much smaller than the one obtained by that rule.
Reduce and to a common denominator.
According to the rule in Art. XIX., the common denominator will be 54, and § = 44 and
It was observed Art. XIX., that the common denominator may be any number, of which all the denominators are factors. 6 and 9 are both factors of 18, therefore they may be both reduced to 18ths. § = 15, and 3=8.
When the fractions consist of small numbers, the least denominator to which the fractions can be reduced, may be easily discovered by trial; but when they are large it is more difficult. It will, therefore, be useful to find a rule for it.
Any number, which is composed of two or more factors, is called a multiple of any one of those factors. Thus 18 is a multiple of 2, or of 3, or of 6, or of 9. It is also a common multiple of these numbers, that is, it may be produced by multiplying either of them by some number. The least common multiple of two or more numbers, is the least number of which they are all factors. 54 is a common multiplę of 6 and 9, but their least common multiple is 18.
The least common denominator of two or more fractions will be the least common multiple of all the denominators; the fractions being previously reduced to their lowest terms. One number will always be a multiple of another, when the former contains all the factors of the latter. 62 × 3, and 9 3 x 3, and 18 2 x 3 x 3. 18 contains the factors 2 and 3 of 6 and 3 and 3 of 9 2 × 3 × 3 × 3