the right hand figure will be even; consequently, it may again be divided by 2; therefore, the whole may be divided by 6 at once.

EXAMPLE.

6) 7416

1236

5. If the last three figures of any number of digits be divisible by 8, the whole is divisible by 8.

Because, when three figures are taken away, and ciphers are substituted, the sum is so many thousands; and as a thousand can be divided by 8, so may any number of thousands; hence, if the last three figures be divisible by 8, it follows that the whole is divisible by 8.

6. When the sum of any number of digits is divisible by 9, the whole is divisible by 9. This is well understood by most arithmeticians, who have attempted to explain it by means of algebraical expressions; yet, as the explanation may not be satisfactory to every one, I will here simply put down a copy of the line from the multiplication table, and make observations upon it.

09 18 27 36 45 54 63 72 81 90

Here, every product is increased by 9, and as 9 is equal to 10 minus 1, consequently the right hand figure of each product in succession is diminished by 1, and the left hand figure (the tens) is increased by 1; therefore the sum of each of their products is the same.

But when several figures are multiplied by 9, the sum of their digits is divisible by 9; and this may be proved in the same manner as laid down in No. 1.

In the above I have put a cipher on the left of the 9, and have taken in the product of nine tens, or 90, merely that the reader may see how the products are inverted at equal distances; for 9 times 1 is 9, or 09; and 9 times 10 is 90; again, 9 times 2 is 18; and 9 times 9 is 81; so that by comparing each product at equal distances, the figures in the

tens' and units' places change their situation with each other; and here the nine digits are expressed only twice with the exception of the 90 (9 times 10), i. e. once in the tens' place, and once in the units' place, in regular order, which cannot be said of any other line in the multiplication table.

7. If the sum of any number of digits be divisible by 9, and it end with a cipher or an even number, it is not only divisible by 9, but by 18; this must be clear from the fact that the product of two odd numbers produces odd; but the product of an odd and even figure produces even: hence, the last right hand figure being even, may again be divided by 2, and this makes good the proposition, that when the sum of any number of digits is divisible by 9, and it ends with an even figure, the whole may be divided by 18; and, if the reader have a ready knowledge of the enlarged multiplication table, it may easily be done in one line.

EXAMPLE. 18) 47502

2639

8. Any number ending with 5 or a cipher, the sum of whose digits is divisible by 3, can be divided by 15, without a remainder; which may be performed in the following

manner.

Subtract from the number its third part, rejecting the cipher in the unit's place, and it will be the quotient.

EXPLANATION OF THE ABOVE.

Let the unit 1 be substituted for either of the above examples; then 1-3 the remainder; and this +10= (for rejecting the cipher in the remainder is dividing it by 10).

9. Any number ending with 25,* 50, 75, or 00, can be divided by 25, without a remainder; and the work can be performed almost instantaneously by the following method:

NOTE.

This No. 25 is not only useful in fractions, but sometimes in the square root, when the figures to be extracted end as above. (See square root.) It is, moreover, useful in the whole of reduction of money, weights, measures, &c. This number, therefore, is of the utmost importance; and should be well attended to whenever an opportunity occurs

See how many times the number 25 is contained in the last 2 figures of each sum; then multiply the remaining figures by 4, because every hundred contains four twentyfives; and carry the number of twenty-fives, and it will give the quotient as in the following examples.

10. Numbers ending with *125, 250, 375, 500, 625, 750, 875, or 000, i. e. with 1, 2, 3, 4, 5, 6, 7 times 125 can be divided by 125, without a remainder.

Rule. See how many times the number 125 is contained in the last 3 figures of each sum, then multiply the remaining figures by 8, observing to carry the number of one hundred and twenty-fives in those figures, and it will give the quotient.

29,425 quot. 14,146 quot. 2,371 quot. 55,588 quot. 11. Numbers ending with 250, 500, 725, or 000, can be divided by 250, without a remainder.

Any numbers may be divided by 25, 125, and 250, by the same rules; but then there will always be a remainder, if

NOTE.

*The No. 125 may also be of great use in the cube root, for when the number whose root is required ends with 125, or any multiple of it, the operation may be considerably shortened. (See cube root.) I have merely mentioned these things here, to lead the reader to observe them as he proceeds.

ending differently to what has been before stated, and such remainder will be what is left as a numerator, having the divisor for a denominator.

CASE FIRST.

To reduce a compound fraction to a single one.

Any number of compound fractions constitute only one fraction, which is called a single fraction when their terms are multiplied together. Thus, when a question is asked, Give me the of of a yard ?" it must be understood, that (1-) of the must be taken from it, and therefore the is made less, which is always the case in compound fractions.

The cancelling of compound fractions before they are brought to a single one is nothing more than reasonable, for it is often better to cancel them in the state they present themselves to notice, than to find their product, and then reduce them; but where there are only two compound fractions it is a matter of indifference, because an ingenious person will multiply the terms, and in the same moment have them in their lowest terms at once, thus, & of

In cancelling where there are many compound fractions, it is advisable to observe, if the product of any two figures in any one of the terms will cancel one or more figures in the other, especially where the figures are not common to the numerator and denominator, because this will save repeated cancelling: for example.

Reduce of of to a single fraction

of of *

Here 4 and 8, the numerators, and 32, the denominator, are cancelled, because 4 times 8 is 32.

Reduce of of to a

single fraction.

Answer, 3.

Reduce of of to a

single fraction.

Answer,

CASE SECOND.

To reduce fractions of different denominations to others that shall have a common denominator.

Rule. Find the least common denominator by inspection (if possible) that can be divided by the original denominators without a remainder.

2. Divide this denominator thus found, by the denominator of each of the original fractions, which quotient, multiplied by each numerator respectively, will be new numerators.

Note. I am aware that the above rule is not new, but yet there are some arithmeticians who multiply the common denominator by the numerator, and divide by the denominator of the original fraction for a new numerator; but it is not advisable. Following the rule above given, will enable the reader to find all the new numerators mentally.

EXAMPLE.

Reduce,, and to common denominator.

Here, to find the common denominator of the above by inspection, we can easily see that no figure will measure either 7 or 11; and although the 8 is a number which can be divided, yet it is impossible to have a less denominator than 616 (the product of 7, 8, and 11.)

Another example, with an additional fraction, say.

,,,

Here it is evident, by inspection, that we may obtain a less common denominator than the product of the whole of them, because the same figure that will measure 8 will also measure 12, namely, 4; for 8 and 12 are in the ratio to one another as 2 to 3; the least common denominator then will be 2 X3 X4 X 7 X 11, or 1848; and therefore the above fractions reduced to a common denominator will be 148, 145, +, and 4.

693

The following is an easy mode of obtaining the smallest multiples, which can scarcely fail of proving a source of great amusement.

If the product of any two denominators be equal to one of the others, reject them, place the remaining denominators in a line, and divide by any figure that will divide two or more of them without a remainder; repeat this process (if necessary) till no two can be divided, (for it is useless to proceed further if two cannot be divided by some figure,) multiply the last quotients into the divisors, and the least common multiple will be obtained.