6. If two different numbers be divisible by any one number, their sum
and their difference will also be divisible by that number.
7. If several different numbers, divisible by 3, be added or multiplied
together, their sum and their product will also be divisible by 3.
8. If two numbers divisible by 9, be added together, their sum of the
figures in the amount will be either 9 or a number divisible by 9.
9. If any number be multiplied by 9, or by any other number divisible
by 9, the amount of the figures of the product will be either 9 or a
number divisible by 9.
10. In every arithmetical progression, if the first and last term be
each multiplied by the number of terms, and the sum of the two products
be divided by 2, the quotient will be the sum of the series.
11. In every geometric progression, if any two terms be multiplied
together, their product will be equal to that term which answers to the
sum of these two indices. Thus, in the series:
1 2 3 4 5
2 4 8 16 32
If the third and fourth terms, 8 and 16, be multiplied together, the
product, 128, will be the seventh term of the series. In like manner,
if the fifth term be multiplied into itself, the product will be the
tenth term; and if that sum be multiplied into itself, the product
will be the twentieth term. Therefore, to find the last, or twentieth
term of a geometric series, it is not necessary to continue the series
beyond a few of the first terms.
Previous to the numerical recreations, we shall here describe certain
mechanical methods of performing arithmetical calculations, such as are
not only in themselves entertaining, but will be found more or less
useful to the young reader.
To Find a Number Thought of.