FACTORS. 98. 1. What is the product when 7 is multiplied by 5? What are the numbers 7 and 5 of their product? 2. What numbers multiplied one by the other will produce 63? What are the numbers 7 and 9 of 63? 3. What are the factors of a number? 4. Name the factors of the following numbers: 20, 36, 45, 48, 60, 35, 72, 50, 21, 40, 24, 32, 64, 81, 80, 56, 44. 5. What numbers will exactly divide 48? 40? 80? 81? 6. Since 6 exactly divides 48, what part of 48 may it be called? 7. Give the exact divisors of the following numbers: 40, 81, 56, 42, 64, 32, 24, 50, 72, 35, 36, 45, 48, 60. 8. What numbers between 0 and 20 have no exact divisors except themselves and 1? What name is given to such numbers? Prime Numbers. 9. What are the prime numbers between 20 and 40? 10. What numbers between 0 and 40 have exact divisors besides themselves and 1? What name is given to such numbers? Composite Numbers. 11. Select from the following numbers, first, the prime numbers: secondly, the composite numbers: 13, 15, 21, 18, 27, 23, 17, 40, 41, 37, 25, 19, 42, 47, 43, 20, 14, 28, 36, 35, 33, 48, 49, 50, 51, 53, 72, 86, 66, 54, 80, 44, 71, 55, 63, 65, 67, 77, 84, 81, 83, 61, 73, 88, 87, 99, 93, 92, 97. 99. A number that expresses whole units is called an Integer or Integral Number. Thus, 18, 25, 30, etc., are integers or integral numbers. 100. The integers which multiplied by one another will produce a number are called the Factors of the number. Thus, 7 and 5 are the factors of 35. 101. An integer that will divide a number without having a remainder is called an Exact Divisor of the number. Thus, 2, 3, 4, and 6 are exact divisors of 12. The factors of a number are exact divisors of it. 102. A number that has no exact divisor except itself and 1 is called a Prime Number. Thus, 1, 3, 5, 7, 11, 13, etc., are prime numbers. 103. A number that has exact divisors besides itself and 1 is called a Composite Number. Hence, a composite number is always the product of two or more factors. Thus, 12, 18, 21, 40, etc., are composite numbers. 104. A number that is exactly divisible by 2 is called an Even Number. Thus, 10, 12, 16, 18, etc., are even numbers. 105. A number that is not exactly divisible by 2 is called an Odd Number. Thus, 3, 5, 9, 11, 13, etc., are odd numbers. TESTS OF DIVISIBILITY. 106. Illustrate with numbers the truth of each of the following statements : 1. Two is an exact divisor of any number whose righthand digit is 0, 2, 4, 6, or 8. 2. Three is an exact divisor of any number, the sum of whose digits is divisible by 3. 3. Four is an exact divisor of a number, if the number expressed by its two right-hand digits is divisible by 4. 4. Five is an exact divisor of any number whose righthand digit is 0 or 5. 5. Six is an exact divisor of any even number, the sum of whose digits is divisible by 3. 6. Eight is an exact divisor of a number, if the number expressed by its three right-hand digits is divisible by 8. 7. Nine is an exact divisor of any number, the sum of whose digits is divisible by 9. 8. Twenty-five is an exact divisor of a number, if the number expressed by its two right-hand digits is divisible by 25. 9. One hundred twenty-five is an exact divisor of a number, if the number expressed by its three right-hand digits is divisible by 125. 10. If an even number is divisible by an odd number, it is divisible by twice that number. 11. An exact divisor of a number is an exact divisor of any number of times that number. 12. An exact divisor of each of two numbers is an exact divisor of their sum and of their difference. EXERCISES. 107. Find by inspection some of the exact divisors of the following numbers: 108. The process of separating a number into its factors is called Factoring. 109. Factors that are prime numbers are called Prime Factors. Thus, 5 and 7 are the prime factors of 35. 110. When numbers have no common factor they are said to be Prime to Each Other. Thus, 7 and 16 are prime to each other, though 16 is not a prime number. 111. The number of times a number is used as a factor is indicated by a small figure called an Exponent. It is written above and at the right of the number. Thus, 5 used as a factor 4 times is indicated by 54. 1. What are the prime factors of 1008? 2 1008 504 222 733 252 EXPLANATION. - Since every prime factor of a number is an exact divisor of the number, the prime factors of 1008 may be found by finding all the prime numbers that are exact divisors of 1008. Since the number is even, 2 is taken for the first prime divisor. Since the quotient is even, 2 is taken for divisor again, and the 9 division is continued until the last quotient is 1. 63 Hence the prime factors are 2, 2, 2, 2, 7, 3, 3 or 24, 7, RULE. - Divide the given number by any prime number that will exactly divide it. Divide this quotient by another prime number, and so continue until the quotient is 1. The several divisors will be the prime factors. What are the prime factors of the following: 112. 1. How many times is 8 times 5 contained in 16 times 5? 4 times 12 in 16 times 12? 5 times 7 in 15 times 7? 2. How many times is 9 x 8 contained in 27 x 8? 8 x 6 in 24 x 6? 15 x 9 in 30 × 9? 4 x 18 in 12 × 18? |