12. Given the product and one of two factors, to find the other factor. 13. Given the multiplicand and multiplier, to find the product. 14. Given the product and multiplicand, to find the multiplier. 15. Given the product and multiplier, to find the multiplicand. 16. Given two numbers, to find their quotient. 17. Given the divisor and dividend, to find the quotient. 18. Given the divisor and quotient, to find the dividend. 19. Given the dividend and quotient, to find the divisor. 20. Given the divisor, quotient, and remainder, to find the dividend. 21. Given the dividend, quotient, and remainder, to find the divisor. 22. Given the final quotient of a continued division and the several divisors, to find the dividend. 23. Given the quotient of a continued division, the first dividend, and all the divisors but one, to find that divisor. 24. Given the dividend and several divisors of a continued division, to find the quotient. 25. Given two or more sets of numbers, to find the difference of their sums. 26. Given two or more sets of factors, to find the sum of their products. 27. Given two or more sets of factors, to find the difference of their products. 28. Given the sum and the difference of two numbers, to find the numbers. PROPERTIES OF NUMBERS 148. 1. What two numbers, besides the number itself and 1, will give a product of 8? 16? 25? 42? 64? 2. What numbers, other than the given number and 1, will exactly divide 9? 15? 36? 48? 55? 3. Of what sets of two numbers is 24 the product? 4. Of what sets of three numbers is 36 the product? 5. What are the smallest numbers, other than 1, that will exactly divide 18? 21? 49? 55? 6. What is the largest number, other than the given number itself, that will exactly divide 22? 24? 30? 40? 7. Name the numbers between 12 and 30, that are the product of two factors greater than 1. Between 30 and 50. 8. Name the numbers between 5 and 20, that have no other factors than the numbers themselves and 1. 9. Of what number are 7 and 8 the factors? 2, 5, and 7? 4, 5, and 3? 2, 3, 5, and 10? DEFINITIONS. 149. The Properties of Numbers are those qualities or elements which necessarily belong to numbers. Numbers are either Integral, Fractional, or Mixed. 150. An Integral Number or Integer is a number representing whole things. (4.) Thus, 8, 23, 30 men, 45 pounds are integral numbers. Integral numbers are either Even or Odd, Prime or Composite. 151. An Even Number is a number that is exactly divisible by 2. All numbers whose unit figure is 0, 2, 4, 6, or 8, are even. 152. An Odd Number is a number that is not exactly divisible by 2. All numbers whose unit figure is 1, 3, 5, 7, or 9, are odd. 153. A Prime Number is a number that has no integral factors except unity and itself. Thus, 2, 3, 5, 11, 23, etc., are prime numbers. 2 is the only even prime number. 154. A Composite Number is a number that has other integral factors besides unity and itself. Thus, 21 is a composite number, since 21-7 × 3. 155. The Factors of a number, are the numbers which multiplied together will produce it. (108.) Thus, 7 and 8 are factors of 56; 3, 4, and 7, of 84. 156. A Prime Factor is a prime number used as a factor. (153.) The prime factors of a number are also the prime divisors of it. 157. An Exact Divisor of a number is one that will divide that number without a remainder. Thus, 6 is an exact divisor of 48, and 9 an exact divisor of 72. 1. The Exact Divisors of a number are also the factors of that number. 2. An exact divisor of a number is sometimes called the measure of that number. 3. When a number is a factor, or divisor, of each of two or more numbers, it is called a common factor, or divisor, of those numbers. 158. Numbers are prime to each other when they have no common integral factors, or divisors. Thus, 9 and 14, 16 and 25 are prime to each other. DIVISIBILITY OF NUMBERS. 159. A number is said to be divisible by another, when there is no remainder after dividing. Any number is divisible 1. By 2, if it is an even number. Thus, 20, 24, 36, and 44 are divisible by 2. 2. By 3, if the sum of its digits is divisible by 3. Thus, 135, 471, and 1134 are divisible by 3. 3. By 4, if its two right-hand figures are ciphers, or express a number divisible by 4. Thus, 300, 432, and 1548 are divisible by 4. 4. By 5, if it ends with a cipher or 5. Thus, 30, 45, and 235 are divisible by 5. 5. By 6, if it is an even number and divisible by 3. Thus, 168, 402, and 1314 are divisible by 6. 6. By 8, if its three right-hand figures are ciphers, or express a number divisible by 8. Thus, 3000, 2728, and 10576 are divisible by 8. 7. By 9, if the sum of its digits is divisible by 9. Thus, 217683 and 401301 are divisible by 9. 8. By 10, if it ends with one or more ciphers. Thus, 40, 500, 3000 are respectively divisible by 10, 100, and 1000. 9. By 7, 11, and 13, if it consists of but four places, the first and fourth being occupied by the same significant figures, and the second and third by ciphers. Thus, 2002, 3003, and 5005, are divisible by 7, 11, and 13. |