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SHORT PROCESSES IN DIVISION.

When there are ciphers at the right of the divisor, the process of division is readily simplified.

The Divisor 1 with Ciphers annexed. 1. Divide 539 by 10. Process.

Explanation. 10 ) 539 Cutting off the digit 9 from the dividend, and the 5319

O from the divisor, we have 53 tens • 1 ten 53,

with 9 remaining. [Principle 6.] Quo. Rem.

RULE. For each cipher in the divisor cut off a digit from the right of the dividend.

2. Divide :

By 10. 1. 6327. 2. 5327. 3. 9732. 4. 9267. 5. 2567.

By 100. 6. 3267. 7. 5327. 8. 9273. 9. 5533. 10. 1234.

By 1000. 11. 6173. 12. 5432. 13. 8650. 14. 3000. 15. 5678.

The Divisor any Significant Figure with Ciphers annexed.

1. Divide 7436 by 3000. Process.

Explanation. 3000 ) 7 436

Cutting off 436 from D. and 000 from d., we have

7 thousands : 3 thousands - 2, with one thousand 2 1436

remaining. 1 thousand + 436 1436. Hence Q. Quo. Rem.

= 2, and R. 1436.

Repeat the principle involved. 2. Divide : 1. 673 by 20.

5. 1074 by 80. 2. 957 by 30.

6. 1096 by 90. 3. 686 by 40.

7. 5736 by 200. 4. 790 by 50.

8. 7300 by 300.

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The Divisor any Number with Ciphers annexed. 1. Divide 5658 by 3200. Process.

Explanation. 3200) 56/58 ( 1 Quo.

Cutting off 58 from D. and 00 from d.,

we have 56 hundreds, quotient, 58 units re32

maining ; 56 hundreds = 32 hundreds = 1, 2458 Rem.

quotient, with 24 hundreds remaining ; 24 hundreds + 58 units 2458, entire remainder.

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REVIEW. 1. Define the following terms : 1. Division.

7. Long Division. 2. Divisor.

8. Short Division. 3. Dividend.

9. Analysis. 4. Quotient.

10. Solution. 5. Remainder.

11. Principle. 6. O as remainder. 12. Parenthesis. 2. What are the principles of division ?

3. In a solution indicated by the signs you have learned to use, in what order is it always safe to use these signs ?

4. Invent five problems whose solution may be indicated by five different signs.

PROPERTIES OF NUMBERS.

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300.

DEFINITIONS AND INDUCTIVE STEPS. 1. A Factor (Latin, “maker”) of a number is one of the numbers which, multiplied together, produce the number, as in 2 X 3 X 4= 24.

3 X
2. Write two factors that will produce 24.

Write four factors 24.

3. Form an equation, putting five factors 4. An Exact Divisor of a number is one of its factors. IV hat are the exact divisor's of 6 ?

5. Since 2 x 3 x 5 30, are 2, 3, and 5 factors of 30, or exact divisors of 30 ?

6. If you have the equation 2 X? = 6, how can you obtain the required factor?

7. Since 2 X 3 X ? 30, how can you obtain the required factor?

8. Then if a number and all its factors are given except

how do we find that one? 9. Has 2, or 3, or 5, any factors except itself and 1 ? 10. A number that has no factors or exact divisors except itself and one is a Prime number, as 2, 3, 5, 7, 11, etc.

11. A number that has factors or exact divisors other than itself and one is a Composite number, as 4, 6, 8, 9, etc

12. The Prime numbers between 1 and 100 are as follows: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

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13. All the other numbers between 1 and 100 are called what? [See 11.]

14. Why are they so named ? Ans.: Because they are composed of factors.

15. 12=3 X 4 is a correct equation. What kind of factor is 3? What kind is 4?

What are the two equal prime factors of 4? Re-write the equation with three prime factors in the second member.

16. An Even number is exactly divisible by 2. 17. An Odd number is not exactly divisible by 2. 18. Is 3 an exact divisor of 12? Of twice 12 ?

Of twice 12? Of three times 12? Of any number of times 12 ?

19. 3 is an exact divisor of 12 and 21. Is it an exact divisor of their sum ? Illustrate.

Is it an exact divisor of their difference? Illustrate.

20. Since numbers are either prime or composite, factors are either prime or composite.

PRINCIPLES. 1. Every composite number is the product of its prime factors.

2. Every prime or composite factor of a number exactly divides that number.

3. Every exact divisor of a number is one of its prime factors or the product of two or more of its prime factors.

4. Every exact divisor of a dividend exactly divides any number of times that dividend.

5. A common divisor of two numbers or dividends exactly divides their sum.

6. A common divisor of two numbers or dividends exactly divides their difference.

7. Any factor of a number becomes a quotient when the number itself becomes a dividend, and its other factor, or the product of its other factors, becomes a divisor.

SUGGESTION.-Pupils should be required to illustrate each of the foregoing principles.

EXERCISES. 1. Write :

1. Three prime numbers exceeding 100.
2. The composite numbers between 75 and 100.
3. An equation, using three composite numbers as

factors.
4. An equation, using three prime numbers as factors.
What kind of number is the second member? Men-

tion the exact divisors of it. 2. Write:

1. Three even numbers.
2. Three odd numbers.
3. The prime numbers between 1 and 50.
4. The prime numbers between 50 and 100.
5. The single even prime number.

6. Three odd numbers that are not prime. 3. Finish the following equations :

1. 1 X 2 X 89 X 97 = ?
2. 3 X 5 X 7 X 31 = ?
3. 47 X ? X 53 = 12,455.
4. ? X1 X 67 X 89 = 59,630.
5. 41 X 43 X 47 X0= ?

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4. Why are not 49, 51, and 63 prime numbers ?

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FACTORING.

Factoring is the process of obtaining the factors or exact divisors of a number. The number factored is, therefore, a dividend.

The most important problem in this connection is to find the prime factors of a number, as a sure means of obtaining certain required divisors or dividends.

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