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Def. 3. A prime number is one which has no measures but unity and itself.

Def. 4. Numbers are prime to each other, when they have no common measure, but unity.

Def. 5. All other than prime numbers are called composite numbers; and prime numbers, which measure them, are called their prime factors.

1.

TO RESOLVE A NUMBER INTO ITS PRIME FACTORS.

Rule. Divide successively by any numbers which measure it, till the quotient is a prime number. The several divisors, if they are primes, or their prime factors, and the last quotient, are the prime factors required.

Obs. 1. Every even number is divisible by 2.

Obs. 2. Every number is divisible by 4, if the number composed of the last two digits be so divisible.

Obs. 3. Every number is divisible by 8, if the number composed of the last three digits be so divisible.

Obs. 4. Every number ending with 1, 2, 3, or more ciphers, is divisible by 10, 100, 1000, &c.

Obs. 5. Every number ending with 5 is divisible by 5.

Obs. 6. A number is divisible by 3 or 9, if the sum of its digits be so divisible.

Obs. 7. A number is divisible by 11, if the difference of the sums of the digits in the odd and even places be so divisible.

EXAMPLE. Resolve 7920 into its prime factors.

7920 = 10X792

= 10X4X198
= 10X4X2X99
= 10X4X2X11X9

= 2X2X2X2X3X3X5X5X11. 11. TO FIND THE GREATEST COMMON MEASURE OF TWO OR MORE

NUMBERS Rule 1. Resolve, if possible, all the numbers into their prime factors; inultiply together all the factors which are common to all the numbers ; the product is the greatest common measure.

Rule 2. Take any two of the numbers; divide the greater by the less ; if there be a remainder, make this now the divisor, and the former divisor the dividend; repeat this process, till there is no remainder. The last divisor is the G. C. M. of the two numbers.

Go through the same process with this G. C. M. and another of the numbers; the G. C. M. of these will be the G. C. M. of the three.

Repeat this process with each number in succession; the last G. C.M. found is the G. C. M. of the whole.

Obs. If any of the numbers be measures of any of the others, those which they measure may be omitted in the work.

EXAMPLES.

1. Find the G. C. M. of 36, 48, 108, 120.

36=2X2X3X3: 48=2X3X2X2X3: 108=2X2X3X3X3 : 120=2X2X2X3X5 ... G. C. M. =2X2X3=12.

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528)1848(3

1584

264)528(2

528

'. G. C. M. of 9768, 11616=264.

264)12408(47

1056

1848
1848

'. G. C. M. of 9768, 11616, and 12408 = 264.

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Def. 1. A multiple of a number is one which contains, or is divisible by, the first, an exact number of times.

Def. 2. A common multiple of two or more numbers is a multiple of each of them; and the least common multiple is the least of such multiples.

TO FIND THE LEAST COMMON MULTIPLE OF TWO OR MORE NUMBERS.

Rule 1. Resolve the numbers, if possible, into their prime factors; write down all the different factors which appear, repeating each the greatest number of times, which it appears in any one of the numbers. The L. C. M. is the product of these factors.

Rule 2. Take any two of the numbers, find their G. C. M., divide one by it, and multiply the other by the quotient; the product is the L. C. M. of the two numbers. Repeat this process with this L. C. M. and a third number; the L. C. M. so found is the L. C. M. of the three. Go through the same process with each number in succession ; the L. C. M. last found is that required.

Obs. If any of the numbers be measures of any of the others, the measures may be omitted in the calculation.

EXAMPLE.
1. Find the L. C. M. of 14, 18, 24, 35, 42, 68.

Omitting 14 as measuring 42; we have
18
24
35 42

68
=3X3X2; =2X2X2X3; =7X5; =2X3X7; =2X2X17.
.:. L. C. M. = 2X2X2X3X3X5X7X17

=360X119=42840.

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Def. 1. Numbers, which are formed by repetition of units, are called integers, or whole numbers.

Def.2. Numbers, which are formed by repetition of equal parts of units, are called fractions.

A fraction may be greater or less than an unit, according as the number of parts taken to form it be greater or less than the number of parts into which the unit is supposed to be divided.

Def.3. A fraction is represented by writing over a line the number of parts taken to form it, and under the line the number which shews into how many parts the unit is divided, or the denomination of the fraetional part. Thus the fraction formed by taking 6 parts of an unit, which is divided into 7 parts, is represented by 4; that formed by taking 13 such parts, by 18. The former fraction, it is evident, is less than one unit; the latter is greater.

Def.4. The figures above the line in a fraction are called the “numerator," those below the line are called the “ denominator."

Def. 5. If the numerator be less than the denominator, the fraction is a “proper fraction:" if greater, an “improper fraction.”

Def. 6. A number partly integral and partly fractional, as 27 is called a "mixed number.”

Def. 7. A fraction of a fraction is called a “compound fraction.”

Def. 8. A“compound expression " is the name given to several quantities connected by the different signs of operations. Those quantities which compose an expression by addition or subtraction are its “terms ;” those which compose an expression by multiplication are its " factors.

Def. 9. A"complex fraction” is one in which numerator and denominator, one or both, are fractions, or fractional expressions.

Def. 10. A decimal fraction is one with the product of any number of tens for its denominator,

Obs. Decimal fractions are called decimals to distinguish them from other fractions called vulgar fractions, or more commonly fractions. Whenever a fraction is spoken of, a vulgar fraction is intended.

A.-Reduction of Fractions. 1. TO REDUCE A FRACTION TO ITS Lowest TERMS. Rule 1. Resolve, if possible, numerator and denominator into their prime factors; and divide by all the common factors.

Rule 2. Divide numerator and denominator by all the common measures, which they may have, in succession.

Rule 3. Divide numerator and denominator by their G. C. M.

Rule 4. If the numerator and denominator be exhibited in the form of the product of several factors, divide any pair of factors in numerator and denominator by any common measure.

When this is done, as far as possible, multiply the remaining factors.

N.B. This last case must be particularly noticed, as containing the principle of " cancelling."

EXAMPLES. 1. Reduce to its lowest terms 1.

25 5X5 IXI 1

175 5X5X7 IXIX7 2. Reduce to its lowest terms 716. 4746)5085(1

339)5085(15 4746

339

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7X11X24X16 4. Reduce to its lowest terms

8X12X35X33 7X11X24X16 IX 1X2X2

4

8X12X35X33 1x1x5X3 15 II. TO REDUCE AN IMPROPER FRACTION TO A MIXED NUMBER,

OR AN INTEGER. Rule. Divide the numerator by the denominator; the quotient is the integer; the remainder is the numerator, and the divisor is the denominator of the fraction, which must be expressed in its lowest terms. Or-Reduce the fraction to its lowest terms, and divide as before.

EXAMPLES. 1. Reduce 57 to a mixed number.

57

3
= 59/9 = 6- = 6
9

3
2. Reduce 49086 to a mixed number.
40086

307)40086(130 = 40086-307.

307
307

938
40086
176

921
= 130-
307
307

176

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Ill. TO REDUCE AN INTEGER TO AN IMPROPER FRACTION WITH

A GIVEN DENOMINATOR. Rule. Multiply the integer by the denominator ; the product is the numerator of the required fraction.

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