EXERCISE 17 Resolve into prime factors and express each number as the product of its prime factors: 1. 8, 12, 16, 18, 20, 24, 27, 28, 30, 32, 36, 39, 40, 42. 2. 45, 48, 49, 50, 56, 60, 65, 69, 72, 75, 77, 80, 84, 88, 92. 3. 98, 99, 111, 117, 119, 120, 124, 128, 132, 133, 135, 140, 144. 4. 240, 720, 343, 512, 216, 729, 736, 608, 544. 5. 1,331, 11,011, 1,309, 858, 1,274, 891, 3,575. 6. Write all the measures of each of the following numbers: 36, 360, 200, 567, 576, 448. 7. Write all the common measures of: (a) 36, 24; (b) 18, 27; (c) 48, 72; (d) 21, 63; (e) 32, 96; (ƒ) 18, 72. When several numbers are to be taken as a whole and made the subject of an operation, they are inclosed in a sign, or symbol, known as a parenthesis, (). Thus, 3 + (9 - 2) signifies that 3 is to be added to the difference of 9 and 2. A number written immediately to the left of a parenthesis denotes multiplication. Thus, 7 × 4 + 3(8 + 5) means 7 times 4 is to be added to 3 times the sum of 8 and 5. A composite number can be resolved into only one set of prime factors. Thus, the prime factors of 36 are 2, 2, 3, 3. 36 = 22 x 32. The product of no other prime numbers will give 36. If a number is prime to each of two other numbers, it is prime to their product. ILLUSTRATION. If 7 is prime to 207 and to 8, then 7 is prime to 8 x 207. For 7 does not appear among the prime factors of the product. Example 1. Find the G. C. M. of 48, 120, 168. Expressing these numbers as products of their prime factors, 48 = 24 x 3. 120 28 x 3 x 5. 2 is contained 3 times as a factor in 168, 3 times as a factor in 120, and 4 times as a factor in 48; 3 is contained once as a factor in each of the numbers. Hence, the G. C. M. = 23 x 3 = 24. To find the G. C. M. of two or more numbers, express each of the numbers as the product of its prime factors, then take the product of the prime factors common to all the numbers, each factor being taken the least number of times it occurs in any of the numbers. LEAST COMMON MULTIPLE Since the L. C. M. of two or more numbers is exactly divisible by each of the numbers, it follows that the L. C. M. contains all the prime factors of each of the given numbers. This fact suggests a method of finding the L. C. M. of two or more numbers. Example. Find the L. C. M. of 48, 60, 72. Any multiple of 48 must contain 2, 4 times as a factor. Any multiple of 72 must contain 3 twice as a factor. Hence, the number 24 x 32 x 5 = 720 contains all the factors of the three numbers 48, 60, 72. Therefore the L. C. M. of 48, 60, 72, is 720. To find the L. C. M. of two or more numbers, resolve each of the numbers into its prime factors, then find the product of all the prime factors of the given numbers, taking each factor the greatest number of times it occurs in any of the numbers. Another Method Example 1. Find the L. C. M. of 48, 60, 72. L. C. M. = 3 × 5 × 2 × 3 × 2 × 2 × 2 = 720. Step 1. Arrange the numbers in a horizontal row. Step 2. Divide by a prime factor common to two or more of the numbers. Set down the quotients and the undivided numbers. Step 3. Treat the second horizontal row in the same manner, and so on until a horizontal row is obtained which contains numbers prime to one another. If, at any stage of the process, a horizontal row contains a number which is a factor of some other number in that row, then strike out such factor. The continued product of the numbers in the last row and of the divisors will be the L. C. M. FRACTIONS If 23 be divided by 4, the process is indicated 22; the quotient is 53. This means 4 is contained in 23, 5 times and 3 remains to be divided by 4. From questions of this character the term fraction (Latin fractus, broken in pieces) has arisen. 22, 53, 4, are all fractions. 3 A Fraction is an indicated Division. For purposes of instruction fractions are regarded from another point of view. If the rectangle ABCD is divided into four equal parts by lines having the same direction as AB, one of these parts is called one fourth of the whole rectangle; two of the parts are called two fourths of the rectangle; three of the parts are called three fourths of the rectangle; and four of the parts are called four fourths of the rectangle. In general, if any one thing is divided into four equal parts, one of the parts is called a fourth; two of the parts are called two fourths; three of the parts, three fourths, etc. Similarly, if any thing is divided into five equal parts, one of the parts is called one fifth; two of the parts are called two fifths; three of the parts, three fifths, etc. D F E In the above rectangle, if the line EF is drawn so as to divide AB and CD each into two equal parts, the whole figure will be broken up into eight rectangles; one of these rectangles is one eighth of the whole; two of them are two eighths; three of them, three eighths, etc. Divide AE and also EB into three equal parts and draw through the points of division lines parallel to BC. What part of ABCD is one of the small rectangles? two of them? etc. D |