Quotient 1584402. 1584402 2 Rem. Rem. 2X8X12+6×12+6=270

4. Divide the product of 36, 17, and 100 by 25.

QUESTIONS IN THE FOUR SIMPLE RULES.

1. Of three bars of iron, one weighs 137 lbs. another 58 lbs. and the third 163 lbs., required the weight of them altogether.

2. A person bought 3008 yards of silk, and sold to one person 1700 yards, and to another 856 yards, how many yards has he left?

3. A person began business with £2000; he cleared every year for 21 years £450, how much was he worth at the end of that time?

4. A person's income is £9125 yearly, how much is that per day?

Def. 1. A measure of a number is another number which divides the first without remainder.

Def. 2. A common measure of two or more numbers is a measure of all of them.

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.

TO FIND THE GREATEST COMMON MEASURE OF TWO OR MORE

NUMBERS.

Rule 1. Resolve, if possible, all the numbers into their prime factors; multiply 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.

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

=3X3X2;

2×2×2×3; =7X5; =2X3X7; =2×2×17.

.. L. C. M. = 2×2×2×3×3×5X7X17

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 fractional part. Thus the fraction formed by taking 6 parts of an unit, which is divided into 7 parts, is represented by ; that formed by taking 13 such parts, by 13. 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."