A number is exactly divisible by 125 if the number formed by its three right-hand digits is divisible by 125. NOTATION 22 is a short way of writing 2 × 2. 23 is a short way of writing 2 × 2 × 2. 24 is a short way of writing 2 × 2 × 2 × 2. 25 is a short way of writing 2 × 2 × 2 × 2 × 2. 26 is a short way of writing 2 × 2 × 2 × 2 × 2 × 2. The result of taking a number any number of times as factor is called a power of the number. 7 x7 = 2401. 2401 is the 4th power of 7. Thus, 747 x 7 x The 4 written to the right of 7 and slightly above it is called the index or exponent of the power. 22 is read 2 square; 23 is read 2 cube, or the cube of 2; 24 is read 2 fourth power, or the fourth power of 2. Example 1. 7)1001 11)143 FINDING PRIME FACTORS Resolve 1001 into its prime factors. We begin with the lowest prime numbers and try them as factors. The units' digit of 1001 is not divisible 13 by 2. This indicates to us that 2 is not a factor of 1001. The sum of its digits is not divisible by 3. This indicates that 3 is not a factor of 1001. Also we see that 5 is not its factor. So we try 7. By actual division we find 7 to be a factor, because it is contained an exact number of times, 143. The next prime number larger than 7 is 11. 11 is contained in 143 (see page 34), and the quotient, 13, is a prime number. Hence, the prime factors of 1001 are 7, 11, 13; and 1001=7 × 11 × 13. Example 2. Resolve 5040 into its prime factors, and express 5040 as the product of prime numbers. Since Divide next Divide by 2 as many times as possible. 315 ends in 5, 5 is a factor of 315. by 3 as often as possible. The prime factors of 5040 are 2, 2, 2, 2, 3, 3, 5, 7. 50402 × 2 × 2 × 2 × 3 × 3 × 5 × 7 = 24 × 32 × 5 × 7. 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. 1331, 11011, 1309, 858, 1274, 891, 3575. 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 added, subtracted, multiplied, or divided like a single number, they may be inclosed in a sign, or symbol, known as a parenthesis. A number written immediately before a parenthesis with no sign between is to be multiplied by the quantity within the parenthesis. Thus, 3(8+5) means 3 times the sum of 8 and 5. Multiplication or division signs are like parentheses in their effect upon the quantities between which they are written. Thus: 3 × 14 + 4 × 11 means the same as (3 × 14) + (4 × 11), which means that 3 times 14 is to be added to 4 x 11. GREATEST COMMON DIVISOR The Greatest Common Divisor is the largest factor that is common to two or more numbers. Thus, 12, 8, and 16 have as common factors both 2 and 4, but 4 is their G. C. D. Example 1. Find the G. C. D. of 48, 120, 168. Expressing these numbers as products of their prime factors, 168 = 2 x 2 x 2 x 3 x7 = 23 × 3 × 7. The common factors are 2, 2, 2, 3. Therefore, the G. C. D. is the product of these, or 23 x 3 = 24. To find the G. C. D. 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. Example 2. Resolve 5040 into its prime factors, and express 5040 as the product of prime numbers. Since Divide next Divide by 2 as many times as possible. 315 ends in 5, 5 is a factor of 315. by 3 as often as possible. The prime factors of 5040 are 2, 2, 2, 2, 3, 3, 5, 7. 5040 = 2 × 2 × 2 × 2 × 3 × 3 × 5 × 7 = 24 x 32 x 5 x 7. 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. 1331, 11011, 1309, 858, 1274, 891, 3575. 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 added, subtracted, multiplied, or divided like a single number, they may be inclosed in a sign, or symbol, known as a parenthesis. A number written immediately before a parenthesis with no sign between is to be multiplied by the quantity within the parenthesis. Thus, 3(8+5) means 3 times the sum of 8 and 5. Multiplication or division signs are like parentheses in their effect upon the quantities between which they are written. Thus: 3 × 14 + 4 × 11 means the same as (3 × 14) + (4 × 11), which means that 3 times 14 is to be added to 4 × 11. GREATEST COMMON DIVISOR The Greatest Common Divisor is the largest factor that is common to two or more numbers. Thus, 12, 8, and 16 have as common factors both 2 and 4, but 4 is their G. C. D. Example 1. Find the G. C. D. of 48, 120, 168. Expressing these numbers as products of their prime factors, 482 x 2 x 2 x 2 x 3 = 1202 x 2 x 2 x 3 x 5 = 168 = 2 × 2 × 2 × 3 × 7 = The common factors are 2, 2, 24 x 3. 23 × 3 × 5. 28 × 3 × 7. 2, 3. Therefore, the G. C. D. is the product of these, or 23 × 3 = 24. To find the G. C. D. 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. |