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. 10)539 5319 Quo. Rem. Explanation. 53, Cutting off the digit 9 from the dividend, and the 0 from the divisor, we have 53 tens 1 ten with 9 remaining. [Principle 6.] RULE. For each cipher in the divisor cut off a digit from the right of the dividend. 3000) 7436 Explanation. Cutting off 436 from D. and 000 from d., we have 7 thousands 3 thousands 2, with one thousand remaining. 1 thousand +436 2 1436 Quo. Rem. 2. Divide: 1. 673 by 20. Repeat the principle involved. 1436. Hence Q. 5. 1074 by 80. 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. 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 × 3 × 4 = 24. 2. Write two factors that will produce 24. Write four factors 24. = 300. 3. Form an equation, putting five factors = 6. If you have the equation 2 x ? 6, how can you obtain the required factor? 7. Since 2 × 3 ×? quired factor? = 30, how can you obtain the re 8. Then if a number and all its factors are given except one, 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. 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 × 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 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. 4. An equation, using three prime numbers as factors. 2. Write: 1. Three even numbers. 2. Three odd numbers. 3. The prime numbers between 1 and 50. 6. Three odd numbers that are not prime. 3. Finish the following equations : 4. Why are not 49, 51, and 63 prime numbers? 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. |