8. A prime number is that which can only be measured by itself, or a unit.

9. That number which is produced by multiplying feveral numbers together is called a compofite number.

10. A perfect number is equal to the fum of all its ali. quot parts.

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 lefs, and this divifor by the remainder, and fo on, always dividing the last divifor by the last remainder, till nothing remain, then will the last divifor 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 ; then of that common measure and one of the other numbers, and fo on, through all the numbers to the laft; then will the greatest common measure, last found, be the answer.

3 If happens to be the common measure, the given numbers are prime to each other, and found to be incommenfurable, or in their lowest terms.

EXAMPLES.

EXAMPLES

1. What is the greatest common measure of 1836,

2. What is the greateft common measure of 1224 and 1080 ?

Anf. 72.

PROBLEM II.

To find the least common multiple of two, or more, numbers.

I.

RULE.

Divide by any number, that will divide two or more of the given numbers, without a remainder, and fet 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 divifors and quotients, will give the multiple required.

EXAMPLES.

EXAMPLES.

1. What is the least common multiple of 6, 10, 16,

X2 X2 X3X4=240 Anf

I furvey my given numbers and find that five will divide two of 4 them, viz 10 and 20, which divide by 5,

2 bringing, into a line with the quotients, the numbers, which 5 will not measure: Again, I view the numbers in the fecond line, and

find 2 will meafure them all, and I get 3, 1, 8, 2, in the third line, and find that 2 will measure 8 and 2, and in the fourth line, get 3, 1, 4, 1, all prime, I then multiply the prime numbers and the divifors continually into each other, for the number fought, and find it to be 240.

2. What is the least common multiple of 6 and 8 ?

Anf. 24

3. What is the least number that 3, 5, 8 and 10 will meafure ? Anf. 120.

4. What is the leaft number which can be divided by the 9 digits feparately, without a remainder ?

Anf. 2520.

REDUCTION OF VULGAR FRACTIONS

Is the bringing of them out of one form into another, in order to prepare them for the operations of Addition, Subtraction, &c.

CASE

CASE I.*

To abbreviate, or reduce Fractions to their lowest terms.

RULE.

Divide the terms of the given fraction by any num. ber, which will divide them without a remainder. and the quotients, again, in the fame manner; and so on, till it appears that there is no number greater than 1,

which

* That dividing both the terms, that is, both numerator and denominator of the fraction, equally by any number, whatever, will give another fraction, equal to the former, is evident : And if those divifions be performed as often as can be done, or the common divifor be the greateft poffible, the terms of the resulting fraction must be the leaft poffible.

NOTE 1. Any number, ending with an even number or cypher, is divisible by 2

2. Any number, ending with 5 or o, is divifible by 5.

3. If the right hand place of any number be o, the whole is divible by 10.

4. If the two right hand figures of any number be divisible by 4, the whole is divisible by 4.

5. If the three right hand figures of any number be divisible by 8, the whole is divifible by 8.

6. If the fum of the digits, conftituting any number, be di visible by 3 or 9. the whole is divisible by 3 or 9

7. If a number cannot be divided by some number less than the fquare root thereof, that number is a prime.

8 All prime numbers, except 2 and 5, have 1, 3, 7, or 9 in the place of units; and all other numbers are composite.

9 When numbers with the fign of Addition or Subtraction between them, are to be divided by ny umber, each of the numbers must be divided: Thus 6+9+12=2+3+4=9.

3

ro But if the numbers have the fign of Multiplication between them; then only one of them must be divided: Thus, 4X6X10_2X6X10_2X6X2 24

-- *24%

which will divide them, and the fraction will be in its lowest terms. Or,

Divide both the terms of the fraction by the greatest common measure, and the quotients will be the terms o the fraction.

192)288(1

192

Therefore 96 is the greatest common meafure.

288

and 96 };!=3 the fame as before.

Multiply the whole number by the denominator of the fraction, and add the numerator of the fraction to the product; under which fubjoin the denominator, and it will form the fraction required.

EXAMPLES.

* All fractions reprefent a divifion of a numerator by the denominator, and are taken altogether as proper and adequate expreffions of the quotient. Thus the quotient of 3, divided 4, is .