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will follow that a unit of any decimal denomination is 1 more
than a multiple of 9.
Illustrations. -

1=0+1
10 = 9+1
100 99 + 1
1000 = 999 + 1

&c., &c. (c.) As a unit of any decimal denomination is 1 more than # multiple of 9, 2 units must be 2 more than such a multiple, 3 units must be 3 more, 4 units 4 more, 5 units 5 more, &c.

Illustrations. Since 1000 =l more than a multiple of 9, 7000 must equal 7 more.

Since 1000000 1 more than a multiple of 9, 7000000 must equal 7 more, &c.

(d.) But the digit figures of a number express the number of units of its various denominations, and therefore any number must be as many more than a multiple of 9 as there are units in the sum of its digit figures. Illustrations. 8235 = 8000 + 200 + 30 + 5.

8000 = 8 more than a multiple of 9.
200 = 2 more than a multiple of 9.
30 = 3 more than a multiple of 9.

5= 5 more than a multiple of 9. Therefore, 8235 = 8 + 2 + 3 + 5, or 18 more than a multiple of 9, or it equals a multiple of 9, plus 18, and must therefore (since 9 is a divisor of 18) be a multiple of 9. Again. 57864 = 50000 + 7000 + 800 + 60 + 4.

50000 5 more than a multiple of 9.
7000 = 7

=8
60 = 6

4 Therefore, 57864 =5+7+8+6 + 4, or 30 more than a multiple of 9, and is therefore (since 9 is not a divisor of 30) not a multiple of 9.

(e.) From these principles it follows, —

1. That every number is equal to a multiple of 9, plus the sum of its digit figures.

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2. That if the sum of the digit figures of any number be subtracted from it, the remainder will be a multiple of 9.

3. That a number is divisible by 9 when the sum of its digit figures is thus divisible.

4. That the remainder obtained by dividing the sum of the digit figures of any number by 9, is the same as that obtained by dividing the number itself by 9.

5. That the difference of any two numbers, the sums of whose digits are alike, will be a multiple of 9.

6. That the divisibility of a number by 9 will not be affected by any change in the order of its digits.

7. That if the digit figures of a number be added together, and then the digit figures of the result, and so on till the sum is expressed by a single figure, that figure will either be 9, or the remainder obtained by dividing the original number by 9. If the figure is 9, the number is a multiple of 9.

Remark. – Finding the excess of any number over a multiple of 9 is called casting out the e’s.

(f.) Consequences of the foregoing. - From the foregoing properties of the number 9, considered in connection with the principles established in 102, come some convenient methods of proving numerical operations ; a few of which we will mention, leaving the pupil to find out the reasons for each.

1. To prove Addition. — Cast out the 9's from the several numbers added, add the results, and cast out the 9's from their sum. Then cast out the 9's from the number obtained as the answer to the question, and if the work be correct, the last two results will be equal.

2. To prove Subtraction. — Cast out the 9's from the subtrahend and remainder, add the results, and cast out the 9's from their sum. Then cast out the 9's from the minuend, and if the work is correct, the last two results will be equal.

3. To prove Multiplication.- Cast out the 9's from the several factors employed, multiply the results together, and cast out the 9's from their product. Then cast out the 's from the product of the original multiplication, and if the work is correct, the last two results will be equal.

4. To prove Division. — Cast out the 9's from the divisor, quotient, and remainder; to the product of the first two results add the last result, and cast out the 9's. Then cast out the 9's from the dividendo and if the work is correct, the last two results will be equal.

(9.) Dependent on the same principles is the following, which the pupil may use with the uninitiated as an arithmetical puzzle.

To tell what figure has been erased. — Tell a person to write any number whatever, without informing you what it is; to subtract the sum of its digit figures from it; to erase from this result any digit figure, other than zero, and write zero in its place; and finally, to add together the digit figures of the number thus obtained, and tell you their sum.

The difference between this sum and the next higher multiple of 9 will show the figure removed. Thus, if the sum is 29, 7 was erased ; if it is 45, 9 was erased ; &c.

Let the pupil explain the reasons of this.
V. Divisibility by 3.

Since 9 is a multiple of 3, every number which is a multiple of 9 must also be a multiple of 3, (Prop. I.) Therefore, a unit of any decimal denomination must be 1 more than a multiple of 3; and hence a number is a multiple of 3 when the sum of its digit figures is such a multiple.

VI. Divisibility by 11.

(a.) A unit of any decimal denomination is either 1 or 10 more than a multiple of 11. Thus,

1=0 X 11 +1

0 X 11 + 10 100 = 9 X 11 + 1 1000 90 X 11 + 10 10000 = 909 X 11+1

100000 = 9090 X 11 + 10 Or, arranging the numbers with reference to the remainders, we have

1=0 X 11 +1
100 9 X 11+
10000 909 X 11+1

1000000 = 90909 X 11 + 1
10=0 X 11 + 10=1 X 11 - 1
1000 = 90 X 11 + 10 = 91 X 11 - 1
100000 = 9090 X 11 + 10= 9091 X 11 - 1

10000000 = 909090 X 11 + 10 = 909091 X 11 - 1 (6.) From which we see that a unit of any denomination expressed by a figure occupying the 1st, 3d, 5th, or any other odd place from the point, is 1 more than a multiple of 11 ; and that a unit of any denomination expressed by a figure

10:

occupying the 2d, 4th, or any other even place from the point, is 1 less than a multiple of 11.

(c.) Hence, on account of the figures occupying odd places from the point, a number is as many more than a multiple of 11 as there are units in the sum of these figures, while on account of the figures occupying even places, it is as many less than a multiple of 11 as there are units in the sum of those figures.

(d.) Hence, every number is equal to some multiple of 11, plus the sum of its digit figures occupying odd places from the point, minus the sum of those occupying even places. If these sums are alike, the additions will equal the subtractions, and the number will be a multiple of 11. If these sums are unlike, their difference will be the excess of the additions over the subtractions, or of the subtractions over the additions.

(e.) If, then, the difference of the sums of the alternate digits is a multiple of 11, the whole number will be either the sum or the difference of two multiples of 11, and hence a multiple of 11; but if this difference is not such a multiple, the whole number will be either the sum or the difference of two numbers, one of which is, and the other is not, a multiple of 11, and hence will not be a multiple of 11.

(f.) Hence, a number is a multiple of 11, when the sums of its alternate digits are equal, or when their difference is a multiple of 11.

First Example. - Is 15873 a multiple of 11?

Solution. The sum of the digits in the odd places is 3 + 8 + 1, or 12; the sum of those in the even places is 7 + 5, or 12. Hence, the two sums are alike, and 15873 is a multiple of 11.

Second Example.- Is 274854 a multiple of 11 ?

Solution. — The sum of the digits in the odd places is 4 + 8+7= 19; the sum of the digits in the even places is 5 + 4 +2=11; the difference between 19 and 11 is 8, which is not a multiple of 11. Hence, 274854 is not a multiple of 11.

VII. If a number is divisible by each of two numbers

which are prime to each other, it will be divisible by their product.

For dividing by one cannot (since the numbers are prime to each other) cast out the other, or any factor of it.

Illustrations. 1. A number which is divisible by 4 and 9 must by divisible by 4 X 9, or 36.

2. A number which is divisible by 8 and 15 must be divisible by 8 X 15, or 120.

3. A number is divisible by 12, when it is by 3 and 4. 4. A number is divisible by 35, when it is by 7 and 5. 5. A number is divisible by 42, when it is by 6 and 7.

VIII. If a number is divisible by each of two numbers which have a common factor, it will not of necessity be divisible by their product.

For dividing by one must cast out the common factor of the two numbers, and if that factor be not taken more than once as a factor of the original number, the quotient will not be divisible by the other of the two numbers.

Illustrations. 84 is divisible by 4 and 6, but not by their product, 24. 72'divisible by 4 and 6, and also by their product, 24.

IX. If one number is not divisible by another, it will not be divisible by any multiple of that other number.

For if a number does not contain once another number, it cannot contain any number of times that other number.

Illustrations. — A number which is not divisible by 2, is not divisible by 4, 6, 8, 10, &c. A number which is not divisible by 3, is not divisible by 6, 9, 12, 15, 18, 21, &c.

X. A number which is divisible by any composite number is divisible by all the factors of that composite number.

For dividing by any composite number is merely dividing by the product of its factors.

104. Recapitulation, for convenience of reference. I. Any number is divisible by 2, 5,31, or any other number which will exactly divide 10, when its right hand figure is thus divisible.

II. Any number is divisible by 4, 20, 25, 50, 121, 162, or any other number which will exactly divide 100, when the number expressed by its two right hand figures is thus divisible.

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