8. A prime number is that, which can only be measured by itself, or an 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 1.

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 laft divifor by the laft 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.

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

EXAMPLES,

EXAMPLES.

1. What is the greatest common measure of 1836,

Therefore, 36 is the anfwer required.

2. What is the greatest common measure of 1224

and 1080 ?

PROBLEM

II.

Anf. 72.

To find the leaf 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 fet the quotients, together with the undivided numbers, in a line beneath.

2. Divide the fecond line, as before, and fo 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 leaft common multiple of 6, 10, 16

5X2 X2 X3 X4 = 240 Anf.

I furvey my given numbers and find that five will divide two of 4 them, viz. 10 and 20, which I divide by 5, 2 bringing, into a line with the quotients, the numbers, which 5 will not measure: Again, 1 view the numbers in the fecond line, and

I

find 2 will measure 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 leaft number that 3, 5, 8 and 10 will measure? 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

To abbreviate, or reduce fractions to their loveft terms.

RULE.

Divide the terms of the given fraction by any number, 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 thofe divifions be performed as often as can be done, or the common divifor be the greatest poffible, the terms of the refulting fraction muft be the leaft poffible.

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

21. 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 be divifible 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 sum of the digits, conftituting any number, be divisible 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 be tween them, are to be divided by any number, each of the numbers must be divided: Thus, 6+9+12=2+3+4=9.

3

IO. 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.

IXI I

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.

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 by 4, is 3.