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PROPERTIES OF THE NUMBER 9.

974. The Properties of the Number Nine are the truths growing out of its relation to the decimal scale.

975. These properties are presented in the following principles:

PRINCIPLES.

1. A number divided by 9 leaves the same remainder as the sum of the digits divided by 9.

Take any num

ber, as 6854, and analyze it as in the margin, and we see that it consists of two parts; the first part a

multiple of 9 and

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..6854=5×9+8×99+6×999 + 4+5+8+6

the second part the sum of the digits. Now the first part is divisible by 9, hence the only remainder that can arise by dividing by 9, must arise from dividing the sum of the digits. Therefore, etc.

2. A number is exactly divisible by 9 when the sum of its digits is divisible by 9.

3. The difference between any number and the sum of its digits is divisible by 9.

4. A number divided by 9 will leave the same remainder if the order of the figures is changed.

5. The difference between two numbers, the sums of whose digits are equal, is exactly divisible by 9.

NOTE.-Pupils should be required to show how the last four principles are derived from the first.

EXAMPLES FOR PRACTICE.

1. Illustrate Principles 1, 2, 3, and 4, with 8704; with 31685.

2. Illustrate Prin. 5 with the numbers 3671 and 5264.

3. If I invert the digits of 74 and subtract the resulting number 47 from 74, the sum of the digits of the remainder will equal 9; explain it.

4. Prove that this is true of any other number of two integral digits in which the difference is a number expressed by two digits.

EXCESS OF 9's.

976. The Excess of 9's in a number is the remainder after dividing it by 9. It is found by the following rule:

Rule. Add the digits, dropping 9 from the sum when this equals or exceeds 9, and thus continue with the excess.

Thus, to find the excess of 9's in 6789, begin at the left and say "6 and 7 are 13, excess 4 and 8 are 12, excess 3 and 9 are 12, excess 3." Find the excess of 9's in

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PROPERTIES OF THE NUMBER 11.

977. The Number 11 has also some peculiar properties which are presented in the following

PRINCIPLES.

1. Every number is a multiple of 11, plus the sum of the digits in the odd places, minus the sum of the digits in the

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see that it consists of

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Multiple of 11.
77+396+5005+59994+

7+5

two parts, the first a multiple of 11, and the second the sum of the digits in the odd places minus the sum of the digits in the even places.

2. A number is exactly divisible by 11 when the sum of the digits in the odd places is equal to the sum of the digits in the even places.

3. A number is exactly divisible by 11 when the difference between the sums of the digits in the odd places and the even places is a multiple of 11.

4. A number increased by the sum of the digits in the even places and diminished by the sum of the digits in the odd places, is exactly divisible by 11.

5. The excess of 11's in any number is not changed by adding any multiple of 11 to the sum of the digits of either order.

EXCESS OF 11's.

978. The Excess of 11's in a number is the remainder after dividing it by 11. It may be found as follows:

Rule. Subtract the term on the left from the next term on the right, the remainder from the next, and thus continue with all the terms, adding 11 to each minuend when less than the subtrahend.

Thus, take the number 24658 and subtract as directed, and the series of remainders will be 4-2, 6-4+2, 5—6+4—2, 8−5+6−4+2= (8+ 6+2)-(5+4), which we see equals the sum of the digits in the odd places minus the sum of the digits in the even places. These remainders can be reduced as we proceed, remembering to add 11 to any minuend when it is less than the subtrahend, which will not affect the (Prin. 5.)

excess.

1. Find the excess of 11's in 273849.

SOLUTION.-2 from 7 leaves 5, 3+11 are 14, 5 from 14 leaves 9, 8+ 11 are 19, 9 from 19 leaves 10, 4+11 are 15, 10 from 15 leaves 5, 5 from 9 leaves 4; hence the excess is 4.

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PROPERTIES OF THE NUMBER 7.

979. The Number Seven has also some peculiar properties, which are presented in the following

PRINCIPLES.

1. Every number is a multiple of 7, plus the sum of the numbers formed by taking its double numerical periods.

Take any number, as 6945391657, and analyze it, as below, and we see that it consists of two parts, the first a multiple of 7, since 999999 is a multiple of 7, and the second the sums of the numbers expressed by the double periods; and this, it will be readily seen, is general.

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2. A number divided by 7 gives the same remainder as the sum of its double numerical periods divided by 7.

3. A number is exactly divisible by 7, when the sum of the numbers expressed by its double numerical periods is divisible by 7.

980. There is another interesting property of the number seven, which is derived in a similar manner.

PRINCIPLES.

1. Every number is a multiple of 7, plus the sum of the odd numerical periods, minus the sum of the even numeri

ical periods.

Take any number, as 6945391657, and analyze it, as in the margin, and we see that

6945391657=

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945× (999999+1)=134999865×7+945 6X (1000000001-1)=857142858×7- 6

(857142858+134999865+55913)×7+945+657—391+6

it consists of two parts; the first is a multiple of 7 and the second the difference between the sums of the odd and the even numerical periods.

2. A number is exactly divisible by 7 when the sum of the odd numerical periods is equal to the sum of the even numerical periods.

3. A number is exactly divisible by 7 when the difference between the sums of the odd periods and the even periods is divisible by 7.

4. A number increased by the sum of the even numerical periods and diminished by the sum of the odd periods, is exactly divisible by 7.

NOTE. Other laws are given in the Philosophy of Arithmetle.

PROOF OF THE FUNDAMENTAL RULES

BY CASTING OUT NINES OR ELEVENS.

981. The Fundamental Rules may be proved by the excess of 9's and 11's.

PROOF OF ADDITION.

982. The Proof of Addition by casting out 9's is based upon the following principle:

Prin. The excess of 9's in the sum of two or more numbers is equal to the excess of 9's in the sum of the excesses of those numbers.

Each number equals a multiple of 9, + the excess; hence their sum will equal a multiple of 9, + the sum of the excesses; consequently the excess of 9's in the sum of the excesses, will equal the excess in the sum of the numbers.

NOTE.-To prove by excess of 11's, proceed as in proving by excess of 9's. Pupils may be required to test each problem by 11 also.

1. Find the sum of 275, 463, and 907, and prove the work.

SOLUTION. The excess of 9's in 275 is 5, in 463 is 4, in 907 is 7, and the excess in the sum of these excesses is 7. The excess in the sum is also 7; hence the work is correct.

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Rule.-I. Find the excess of 9's in each number, then the excess in the sum of these excesses, and then the excess in the sum of the numbers.

II. If the work is correct, the last two excesses will be equal.

NOTES.-1. We need not write the excess of each number, but can pass from one number to another and write the last excess. We can also add in columns for excess, as well as in rows.

2. This method fails when the digits are misplaced, or when one digit is as much too great as another is too small.

Add and prove the following:

2. 6573+8325+5641+4319+3978+6807.
3. 5432+6431+27944+56352+78698.

4. 46932+79876+85432+65435+57697.
5. 443367637389+457934+697989+609687.

PROOF OF SUBTRACTION.

983. The Proof of Subtraction by casting out 9's is based upon the following principle:

Prin. The excess of 9's in the minuend equals the excess of 9's in the sum of the excesses of the subtrahend and remainder

This is evident from the principle in the previous case, since the minuend equals the sum of the subtrahend and remainder.

OPERATION.

1. Subtract 2562 from 4625, and prove the work. SOLUTION.-The excess of 9's in the minuend is 8, in the subtrahend 6, in the remainder 2, and the excess in the sum of the excesses of the subtrahend and remainder is 2+6, or 8, the same as the excess of the minuend; hence the work is correct.

4625 excess 8 2562

66

6

2063 excess 2

Rule.-I. Find the excess of 9's in each of the three terms, and the excess in the sum of the excesses of the subtrahend and remainder.

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