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7. What is the least common multiple of 365, 910, 2217, and 2424 ?
11. We may also find the least common multiple of two or more numbers by the following
RULE.* Write the numbers in a horizontal line, divide them by any prime number, which will divide two or more of them, place the quotients with the undivided terms for a second horizontal line, proceed with this second line as with the first ; and so con. tinue until there are no two terms which can be divided. The
* This rule is usually given as follows: "Write down the numbers in a line, and divide them by any number that will measure two or more of them; and write the quotients and undivided numbers in a line be. neath. Divide this line as before, and so on, until there are no two numbers that can be measured by the same divisor ; then the continual product of all the divisors and numbers in the last line will be the least common multiple required.”
The above we have copied from Mr. Adams' Arithmetic; nearly all our arithmetics give in substance the same rule. We will now show by an example that this rule may give very different results, depending upon the divisors used, and of course the rule is in fault.
EXAMPLE. What is the least common multiple of 12, 16, and 24? We will work this example in three ways, as follows. 1st OPERATION.
2d OPERATION. 3d OPERATION. 12112, 16, 24
8112, 16, 24 4/12, 16, 24 2 1, 16, 2 312, 2, 3
3 3, 4, 6 11, 8, I 24, 2, 1 21, 4, 2
2, 1, 1 11, 2, I 12 X 2 X8=192 8X3 X2 X2=96 4X3X2 X2=48
These operations, which are wrought strictly by this rule, give 192, 96, and 48 for the least multiple of 12, 16, and 24. Hence the rule is wrong, and can not be depended upon. The least common multiple of 12, 16, and 24, is truly 18, as may be found by either of our rules.
continued product of the divisoss and the numbers in the last horizontal line will give the least common multiple.
Examples 1. What is the least common multiple of 28, 35, 42, 77, and 70 ?
12, 1, 3, 11, 1. Hence, 7x5X2X2 X3x11=4620, is the multiple sought.
2. What is the least common multiple of 46, 92, 374, and 23 ?
Ans. 17204. 3. What is the least common multiple of 5, 15, 36, and 72 ?
Ans. 360. 4. What is the least common multiple of 11, 77, 88, and 92 ?
5. What is the least common multiple of 14, 51, 102, and 500 ?
12. Suppose we wish to find all the divisors of 36, we proceed as follows: We resolve 36 into its prime factors, and thus obtain 36=22 X 33.
Now it is obvious, that any combination of 2 and 3, which does not make use of these factors in a higher power than they occur in 2' x 32 must be a divisor of 36. All such combinations can be found by multiplying 1+2+4 by 1+3 +9, performing this multiplication we obtain
i+2+4+3+6+12+9+18+36. Therefore, the divisors of 36 are 1, 2, 4, 3, 6, 12, 9, 18, and 36. Hence, to find all the divisors of any number, we have this
RULE. Resolve the number into its prime factors, form as many se. ries of terms as there are prime factors, by making 1 the first term of any one of the series, the first power of one of the prime factors for the second term, the second power of this factor for the third term, and so on, until we reach a power as high as occurred in the decomposition. Then multiply these series to. gether, (by rule under Art. 4,) and the partial products thus obtained will be the divisors sought.
Examples. 1. What are the divisors of 48 ?
Here we find 48=24 x3. Therefore our series of terms will be 1+2+4+8+16 and 1+3, multiplying these togeth. er (by rule under Art. 4,) we get
1+2+4+5+16+3+6+12+24+48. Therefore, the divisors of 48 are 1, 2, 4, 8, 16, 3, 6, 12, 24, and 48. 2. What are the divisors of 360 ?
S 1, 2, 4, 8, 3, 6, 12, 24, 9, 18, 36, 72,
* 5, 10, 20, 40, 15, 30, 60, 120, 45, 90, 180, 360. 3. What are the divisors of 100 ?
Ans. 1, 2, 4, 5, 10, 20, 25, 50, 100. 4. What are the divisors of. 810?. 5. What are the divisors of 920 ? 6. What are the divisors of 840?
13. Since the series of terms, which we multiplied to. gether by the last rule, to obtain the divisors of any number, commenced with 1, it follows that the number of terms in each series will be one more than the units in the exponent of the factor used.
Hence, to find the number of divisors of any number, without exhibiting them, we have this
Resolve the number into its prime factors, increase the exponents by a unit, and then take their continued product, and it will express the number of divisors.
: Example. 1. How many divisors has 4320?
4320=25 X 33 X5. In this case the exponents are 5, 3, and 1, each of which being increased by one, we obtain 6, 4, and 2, the continued product of which is 6 X 4x2=48, the number of divisors sought. 2. How many divisors has 300 ?
Ans. 18. 3. How many divisors has 3500 ?
Ans. 24. 4. How many divisors has 162000?
Ans. 100. 5. How many divisors has 824 ?" 6. How many divisors has 1172 ? 17. How many divisors has 6336?
FRACTIONS. 14. A fraction is an expression representing a part of a unit.
VULGAR FRACTIONS. 15. A vulgar fraction consists of two numbers, the one placed above the other as in division. '
The number above the line is called the numerator ; the number below the line is called the denominator.
Thus, is a vulgar fraction, whose numerator is 5, and whose denominator is 8: it is read five-eighths.
The denominator shows how many parts the unit is divided into ; and the numerator shows how many of these parts are used.
Thus, denotes that the unit is divided into 8 equal parts, and that 5 of these parts are used.
When the numerator is equal to the denominator the fraction is equivalent to a unit. Thus, , ii, , and 31, are each equivalent to 1.
When the numerator is less than the denominator, the value of the fraction is less than a unit; it is then called a proper fraction.
Thus, , 4, 1f, and y, are each proper fractions.
When the numerator is larger than the denominator, its value is then more than a unit; it is therefore called an improper fraction.