1. Resolve 1961 into its prime factors. 1961 1961 OPERATION. = 37= = 53 ANALYSIS. Cutting off the two right hand figures of the given number, and referring to the table, column No., we find the other part, 19, in bold-face type; and under it, in the same division of the column, we find 61, the figures cut off; at the right of 61, in column Fac., we find 37, the least prime factor of the given number. Dividing by 37, we obtain 53, the other factor. 2. Resolve 188139 into its prime factors. EXAMPLES FOR PRACTICE. 1. Resolve 18902 into its prime factors. 2. Resolve 352002 into its prime factors. 3. Resolve 6851 into its prime factors. 4. Resolve 9367 into its prime factors. 5. Resolve 203566 into its prime factors. 6. Resolve 59843 into its prime factors. 7. Resolve 9991 into its prime factors. 8. Resolve 123015 into its prime factors. 9. Resolve 893235 into its prime factors. 10. Resolve 390976 into its prime factors. 11. Resolve 225071 into its prime factors. 12. Resolve 81770 into its prime factors. 13. Resolve 6409 into its prime factors. 14. Resolve 178296 into its prime factors. 15. Resolve 714210 into its prime factors. Ans 2, 13, 727. CASE II. 145. To find all the exact divisors of a number. It is evident that all the prime factors of a number, together with all the possible combinations of those prime factors, will constitute all the exact divisors of that number, (142, II). 1. What are all the exact divisors of 360? ANALYSIS. By Case I we find the prime factors of 360 to be 1, 2. 2, 2, 3, 3, and 5. As 2 occurs three times as a factor, the different combinations of 1 and 2 by which 360 is divisible will be 1, 1×2=2, 1 × 2 × 2=4, and 1 × 2 × 2 × 2=8; these we write in the first line. Multiplying the first line by 3 and writing the products in the second line, and the second line by 3, writing the products in the third line, we have in the first, second and third lines all the different combina tions of 1, 2, and 3, by which 360 is divisible. Multiplying the first, second and third lines by 5, and writing the products in the fourth, fifth and sixth lines, respectively, we have in the six lines together, every combination of the prime factors by which the given number, 360, is divisible. Hence the following RULE. I. Resolve the given number into its prime factors. II Form a series having 1 for the first term, that prime factor which occurs the greatest number of times in the given number for the second term, the square of this factor for the third term, and so on, till a term is reached containing this factor as many times as it occurs in the given number. III. Multiply the numbers in this line by another factor, ind these results by the same factor, and so on, as many times as this factor occurs in the given number. IV. Multiply all the combinations now obtained by another factor in continued multiplication, and thus proceed till all the dif ferent factors have been used. ALL the combinations obtained will be the exact divisors sought. EXAMPLES FOR PRACTICE. 1. What are all the exact divisors of 120? Ans. 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120. 2. Find all the exact divisors of 84. Ans. 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84. 3. Find all the exact divisors of 100. Ans. 1, 2, 4, 5, 10, 20, 25, 50, 100. 4. Find all the exact divisors of 420. Ans. ( 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420. 5. Find all the exact divisors of 1050. Ans. ( 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 25, 30, 35, 42, 50, 70, 75, 105, 150, 175, 210, 350, 525, 1050. GREATEST COMMON DIVISOR. 146. A Common Divisor of two or more numbers is a number that will exactly divide each of them. 147. The Greatest Common Divisor of two or more numbers is the greatest number that will exactly divide each of them. 148. Numbers Prime to each other are such as have no common divisor. NOTE. A common divisor is sometimes called a Common Measure; and the greatest common divisor, the Greatest Common Measure. CASE I. 149. When the numbers can be readily factored. It is evident that if several numbers have a common divisor, they may all be divided by any component factor of this divisor, and the resulting quotients by another component factor, and so till all the component factors have been used. on, |