Subtract and test : 10. 964971 - 346782 11. 786482 – 648765 12. 870006 – 738104 13. 806464 - 309874 14. 684671 - 346721 MULTIPLICATION AND DIVISION Multiplication is the process of taking one number as many times as there are units in another. The multiplicand is the number multiplied. A concrete number is a number that refers to some particular object; as, $8, 4 books. An abstract number is a number that does not refer to some particular object; as 8, 4. The multiplier must always be regarded as an abstract number. The product and multiplicand are like numbers. Multiply and test: 1. 6425 : Ca. 245 Form 100 problems by multi2. 1024 b. 344 plying each multiplicand by 3. 8720 c. 564 each of the multipliers, thus : 4. 9652 1 a. 245 x 6425= ? 5. 8665 e. 804 1 b. 344 x 6425 = ? 6. 7894 if. 961 8 . 968 x 7695 = ? 7. 8465 g. 869 Write, solve, and test each 8. 7695 h. 796 problem in 13 minutes. 9. 8425 i. 968 10. 9476 13. 898 *Division is the process of finding how many times one number contains another or of separating a number into equal parts. The separation of a number into equal parts is also called partition. Thus,$ 250 = 5, or } of $250, is an example in partition. The remainder is the part of the dividend remaining when the division is not exact. The sign → indicates division. Division is also indicated as follows: 35 divided by 7 may be written 36 or 7)35. If both dividend and divisor are concrete, they must be like numbers, and the quotient is abstract. If the dividend is concrete and the divisor is abstract, the dividend and quotient are like numbers. Thus,$27 = $3 = 9; but $27 = 3 = $9. Divide and test : 1. 6464341 a. 268 Form 100 problems by divid. 2. 7846760 b. 354 ing each dividend by each of 5864548 C. 676 the divisors, thus : 4. 8645341 d. 758 1 a. 6464341 - 268 = ? 9624872 e. 865 16. 6464341 : 354 = ? 6. 7784100 984 6 c. 7784100 = 676 = ? 7. 6810404 g. 789 Write, solve, and test each 8. 7904025 h. 897 problem in 2 minutes. 9. 4867045 ¿. 509 LO. 3234567 1 ;. 890 FACTORS AND DIVISORS 1. What two numbers will give 6 as a product? 8 as a product? 10 as a product? 2. What are 2 and 3 in relation to 6? 4 and 2 in relation to 8? 5 and 2 in relation to 10? An integer or an integral number is a whole number. The factors of a number are the integers whose product is the number; thus, 5 and 2 are factors of 10. 3. Name two factors that produce 24, 32, 40, 56, 49, 72, 96. A factor of a number is an exact divisor of the number ; that is, it is contained in the product an integral number of times. 4. Name the exact divisors of 54, 81, 48, 36, 66, 64, 63. 2 x 2 = 4 5. Observe the two equal factors that pro 3 x 3 = 9 duce 4; 9; 16. 4 x 4 = 16 2 x 2 x 2 = 8 6. Observe the three equal factors that pro 3 x 3 x 3 = 27 duce 8; 27; 64. 4x4 x 4 = 64 . Instead of repeating a factor, a small figure called an exponent may be written to the right and a little above the number to show how often it is used as a factor ; thus, 33 = 3 x 3 x 3 = 27; 24 = 2 * 2 * 2 x 2 = 16. 7. What number will divide 9 and 10? 21 and 25? Numbers are prime to each other when they have no common factor ; thus, 9 and 10 are prime to each other. Even numbers are numbers that contain the factor 2. 8. What are the factors of 7? of 11? Observe that 7 and 11 have no exact divisors except themselves and one. A prime number is one that has no exact divisor except itself and one ; thus, 5, 2, and 3 are prime numbers. 9. Name all the prime numbers to 31. 10. What are the factors of 15? Observe that 15 can be divided by 3 and 5. It is composed of other factors than itself and one. A composite number is one that has other exact divisors than itself and one; thus, 6 and 10 are composite numbers. 11. Name all the composite numbers to 50. TESTS OF DIVISIBILITY 1. Divide 12, 24, 26, 38, and 50 each by 2. What is the ones' figure in each of the dividends? Divide other numbers ending in 2, 4, 6, 8, or 0 by 2. A number is divisible by 2, if the ones' figure is 2, 4, 6, 8, or 0. 2. Divide 15, 25, 40, 125, 150 each by 5. What is the ones? figure in each dividend ? Divide other numbers end ing in 5 or 0 by 5. • A number is divisible by 5, if its ones' figure is 5 or 0. 3. Divide 36, 69, 48, 72, 162, 369 each by. 3. Notice that the sum of the digits (that is, of the figures) in each number is divisible by 3. Divide by 3, other numbers the sum of whose digits is divisible by 3. • A number is divisible by 3, if the sum of its digits is divisible by 3. |