6. 5 is a factor of every number whose right-hand figure is either 5 or 0; as, 15, 20, &c. 7. 6 is a factor of every even number that 3 will exactly divide; thus, 24, 108, and 360 have each 6 as a factor. 8. 7 is a factor of every number occupying four places whose two righthand figures are contained in the left-hand ligure or figures exactly 3 times ; thus, 2107 and 3913 have cach 7 as a factor. 9. 7 is a factor of every number occupying three or four places, when the two right-hand figures contain the left-hand figure or tigures exactly 5 times; thus, 840, 945, and 1155 have each 7 as a factor. 10. 8 is a factor of every number whose three right-hand figures 8 will exactly divide ; thus, 5072, 11240, and 17128 have each 8 as a factor. 11. 9 is a factor of every number the sum of whose digits 9 will exactly divide; thus, 27, 432, and 20304 have each 9 as a factor. 12. 10 is a factor of every number whose right-hand figure is 0; as, 20, 30, &c. 13. 7, 11, and 13 are factors of any number occupying four places in which two like figures have two ciphers between them ; as, 3003, 4004, 9009, &c. 14. Every prime number, except 2 and 5, has 1, 3, 7, or 9 for the righthand figure. OPERATION. 2 1 2 4 114. To find the prime factors of numbers. Ex. 1. Find the prime factors of 24. Ans. 2, 2, 2, 3. We divide by 2, the least prime number greater than 1, and obtain the quotient 12. And since 12 is a composite number; we divide this 2 1 2 also by 2, and obtain a quotient 6. We divide 2 6 6 by 2, and obtain 3 for a quotient, which is a prime number. The several divisors and the 3 last quotient, all being prime, constitute all the prime factors of 24, which, multiplied together, equal 2 X 2 X 2 X 3 = 24. RULE. - Divide the given number by any prime number, greater than 1, that will divide it, and the quotient, if a composite number, in the same manner; and continue dividing until a prime number is obtained for a quotient. The several divisors and the last quotient will be the prime factors required. Note. — The composite factors of any number may be found by multiplying together two or more of its prime factors. 113. Of what numbers is 5 a factor? Of what is 6 a factor? Of what is 7 a factor? Of what is 8 a factor? Of what is 9 a factor? What is the right-hand figure of every prime number ? — 114. The rule for finding the prime factors of numbers ? How may the composite factors of numbers be found? EXAMPLES FOR PRACTICE. 2. What are the prime factors of 36? Ans. 2, 2, 3, 3. 3. What are the prime factors of 48 ? Ans. 2, 2, 2, 2, 3. 4. What are the prime factors of 56 ? Ans. 2, 2, 2, 7. 5. What are the prime factors of 144? Ans. 2, 2, 2, 2, 3, 3. 6. Find the prime factors of 3420 ? Ans. 2, 2, 3, 3, 5, 19. 7. What are the prime factors of 18500 ? Ans. 2, 2, 5, 5, 5, 37. 8. What are the prime factors of 19965 ? Ans. 3, 5, 11, 11, 11. 9. What are the prime factors of 12496 ? Ans. 2, 2, 2, 2, 11, 71. 10. What are the prime factors of 17199 ? Ans. 3, 3, 3, 7, 7, 13. 11. What are the prime factors of 7800 ? Ans. 2, 2, 2, 3, 5, 5, 13. CANCELLATION. 115. If the dividend and divisor are both divided by the same number, the quotient is not changed. Thus, if the dividend is 20 and the divisor 4, the quotient will be 5. Now, if we divide the dividend and divisor by some number, as 2, we obtain 10 and 2 respectively; and 10 = 2=5, the same as the original quotient. Also, if the dividend and divisor are both multiplied by the same number, the quotient is not changed. 116. If a factor in any number is canceled, the number is divided by that factor. Thus, if 15 is the dividend and 5 the divisor, the quotient will be 3. Now, since the divisor and quotient are the two factors, which, being multiplied together, produce the dividend (Art. 50), if we cross out or cancel the factor 5, the remaining 3 is the quotient, and by the operation the dividend 15 has been divided by 5. 117. Cancellation is the method of shortening arithmetical operations by rejecting any factor or factors common to the divisor, and dividend. 115. What is the effect on the quotient when the dividend and divisor are divided by the same number? - 116. The effect of canceling a factor of any number? - 117. What is cancellation ? Ex. 1. A man sold 25 hundred weight of iron at 5 dollars per hundred weight, and expended the money for flour at 5 dollars per barrel; how many barrels did he purchase? Ans. 25 barrels. OPERATION. We first indicate by their signs the Dividend 5 x 25 multiplication and division required by 25. the question. Then, observing 5 to be Divisor $ a cominon factor of the divisor and dividend, we divide the divisor and dividend by this factor, or, which is the same thing, cancel or reject it in both, and obtain 25 for the quotient. 2. Divide the product of 12, 7, and 5 by the product of 5, 4, and 2. Ans. 103 OPERATION. 3 2 1 = 104, Quotient. 2 Finding 4 in the divisor to be a factor of 12 in the dividend, we divide 12 by 4, canceling these numbers, and use the 3 instead of 12. The factor 5, common to both dividend and divisor, having been canceled, we divide the product of the remaining factors in the dividend by the product of those in the divisor, and obtain the quotient 104. 3. Divide the product of 8, 5, 16, and 21 by the product of 10, 4, 12, and 7. OPERATION. 4 Dividend 8 X $ X 16 X 2 1 = 4, Quotient. Divisor X X 4 X X 2 X 7 The product of the factors 8 and 5 in the dividend is equal to the product of 10 and 4 in the divisor; therefore we cancel these factors. Finding 16 in the dividend and 12 in the divisor may be exactly divided by 4, they are canceled, and use made of the quotients. Again, as the product of the factors 3 and 7 of the divisor equals the 21 of the dividend, we cancel the 3, 7, and 21. The factor 4 alone remaining is the quotient. 117. How do you arrange the dividend and divisor for cancellation? How do you then proceed? Is the factor 5, in Ex. 1, reduced to 0 or 1 by being canceled ? How do you proceed when a number in the dividend and another in the divisor have a common factor ? How do you proceed when the products of two or more factors in the dividend and divisor are alike? RULE. Cancel the factor or factors common to the dividend and divisor, anıl then dicule the product of the fucior's remaining in the dividend by the product of those remaining in the divisor. NoТЕ Т. In arranging the numbers for canceilition, the dividend may be written above the livisor with a horizontal we between them, as in division (Art. +7); or, as some prefir, the dividend may be written on the right of divisor, with a vertical line between them. Note 2. — Canceling a factor does not leave 0, but the quotient 1, to take its place, since rejecting a factor is the same as dividing by that factor (Art. 116). Therefore, for every factor canceled I remains. EXAMPLES FOR PRACTICE. 4. Divide 42 X 19 by 19. Ans. 42. 5. Divide the product of 8, 6, and 3, by the product of 6, 3, and 4. Ans. 2. 6. Divide the product of 17, 6, and 2, by the product of 6, 2, and 17. Ans. 1. 7. Sold 15 pieces of shirting, and in each piece there were 30 yards, for which I received 10 cents per yard; expended the money for 10 pieces of calico, each containing 15 yards ; what was the calico per yard ? Ans. 30 cents. 8. Divide the product of 12, 7, and 5, by the product of 2, 4, and 3. 9. Divide the product of 20, 13, and 9, by the product of 13, 16, and 1. Ans. 114 10. Divide the product of 9, 8, 2, and 14, by the product of 3, 4, 6, and 7. Ans. 4. 11. Divide the product of 16, 5, 10, and 18, by the product of 8, 6, 2, and 12. Ans. 124 12. Divide the product of 22, 9, 12, and 5, by the product of 3, 11, 6, and 4. Ans. 15. 13. Divide the product of 25, 7, 14, and 36, by the product of 4, 10, 21, and 54. Ans. 115 14. Divide the product of 26, 72, 81, and 12, by the product of 36, 13, 24, and 54. Ans. 3. 15. Divide the product of 8, 5, 3, 16, and 28, by the product of 10, 4, 12, 4, and 7. Ans. 4. 16. Divide the product of 8, 4, 9, 2, 12, 16, and 5, by the product of 4, 6, 6, 3, 8, 4, and 20. Ans. 2. 17. Divide the product of 6, 15, 16, 24, 12, 21, and 27, by the product of 2, 10, 9, 8, 36, 7, and 81. Ans. 174 Ans. 8. 117. The rule for cancellation ? How may the numbers be arranged for canceling? What remains for every factor canceled ? A COMMON DIVISOR. 118. A Common Divisor of two or more numbers is any number that will divide them without a remainder ; thus, 2 is a common divisor of 2, 4, 6, and 8. 119. To find a common divisor. Ex. 1. What is the common divisor of 10, 15, and 25? Ans. 5. OPERATION. We resolve each of the given numbers into two 10 5 X 2 factors, one of which is common to all of them. 15 5 X 3 In the operation 5 is the common factor, and there25 5 X 5 fore must be a common divisor of the numbers. RULE. - Resolve each of the given numbers into two factors, one of which is common to all of them, and this common factor is a common divisor. EXAMPLES FOR PRACTICE. 2. What is the common divisor of 3, 9, 18, 24 ? Ans. 3. 3. What is the common divisor of 4, 12, 16, 28 ? Ans. 2 or 4. 120. A divisor of any factor of a number is a divisor of the number itself. Thus 3, a divisor of 9, a factor of 45, is a divisor of 45 itself. 121. A common divisor of two numbers is a divisor of their sum and of their difference. Thus 4, a common divisor of 16 and 12, is a divisor of their sum, 28, and of their difference, 4. 122. A common divisor of the remainder and the divisor is a divisor of the dividend. Thus, in a division having 12 for remainder, 36 for divisor, and 48 for dividend, 12, a common divisor of the 12 and the 36, is also a divisor of the 48. THE GREATEST COMMON DIVISOR. 123. The Greatest Common Divisor of two or more nbers is the greatest number that will divide each of them without a remainder. Thus 6 is the greatest common divisor of 12, 18, and 24. 118. What is a common divisor of two or more numbers ? — 119. The rule?-121. Of what is the co'nmon divisor of two numbers a divisor?. 122. Of what is a common divisor of the less of two numbers and of their difference a divisor ? — 123. What is the greatest common divisor of two oi more numbers ? |