Divisibility of Numbers. 164. PROBLEMS. 1. How many times is 25 contained in 2450 ? * 2. How many times is 25 contained in 3775? 3. How many times is 33} contained in 4666? 4. How many times is 332 contained in 34331? 5. How many times is 121 contained in 47374? I 6. How many times is 123 contained in 36621? 7. How many times is 16 contained in 25333 ? 8. How many times is 16% contained in 4550 ? 165. Numbers exactly divisible by 9. 1. Any number is exactly divisible by 9 if the sum of its digits is exactly divisible by 9. EXPLANATORY NOTE. — Any number more than nine is a certain number of nines and as many over as the number indicated by the sum of its digits. Thus, 20 is two nines and 2 over ; 41 is four nines and 4+1 over; 42 is four nines and 4+2 over; 200 is twenty-two nines and 2 over; 300 is thirty-three nines and 3 over ; 320 is a certain number of nines and 3+2 over ; 321 is a certain number of nines and 3 + 2 +1 over. 326 is a certain number of nines and 3+2+6 over ; but 3 + 2 + 6 = 11, or another nine and 2 over. 2. Read the Explanatory Note” carefully and tell which of the following are exactly divisible by 9: 3256, 4266, 2314, 2574. 166. PROBLEMS. over. 1. 4625 is a certain number of 9's and 2. 3526 is a certain number of 9's and 3. 2154 is a certain number of 9's and over. over. * 25 is contained in 2450 (4 X 24) + 2 times. Why? 124 is contained in 4737(8 x 47) + 3 times. Why? Divisibility of Numbers. 1. Any integral number is exactly divisible by each of its prime factors and by the product of any two or more of its prime factors. Thus, 30, (2 x 3 x 5), is exactly divisible by 2, by 3, by 5, and by (2 x 3), 6, and by (2 x 5), 10, and by (3 x 5), 15. 2. The exact integral divisors of 36, (2 x 2 x 3 x 3), are and 2, 3, 168. PRIME FACTORS, COMMON DIVISORS, AND GREATEST COMMON DIVISORS. 1. Any prime factor or any product of two or more prime factors common to two or more numbers is a common divisor of the numbers. Thus, the numbers 30,(2 x 3 x 5), and 40, (2x2x2x5), have the factors 2 and 5 in common. So the common divisors of 30 and 40 are 2, 5, and 10, and the greatest common divisor is 10. RULE. To find the greatest common divisor of two or more numbers, find the product of the prime factors common to the numbers. 2. Find the g. c. d. of 50, 75, and 125.* OPERATION No. 1. OPERATION No. 2. 50 = 2 x 5 x 5. 5 50 75 125 75 = 3 x 5 x 5. 5 10 15 25 125 = 5 x 5 x 5. 2 3 5 5 x 5 = 25, g. c. d. 5. Find the g. c. d. of 64, 96, 256. * The letters g. c. d. are sometimes used instead of the words greatest common divisor. Divisibility of Numbers. 6. Find the g. c. d. of 610 and 760. Operation. Explanation. 640)760(1 The number 760 is an integral number 640 of times the g. c. d., whatever that may be ; 120)640(5 so is the number 640. We make an incom600 plete division of 760 by 640 and have as a remainder the number 120. Since 640 and 40)120(3 120 760 are each an integral number of times the g. c. d., their difference, 120, must be an integral number of times the g. c. d. : for, taking an integral number of times a thing from an integral number of times a thing, must leave an integral number of times the thing. Therefore, no number greater than 120 can be the g. c. d. But if 120 is an exact isor of 640, it is also an exact divisor of 760, for it will be contained one more time in 760 than in 610. We make the trial and find that 120 is not an exact divisor of 640; there is a remainder of 40. Since 600, (120 x 5), and 640 are each an integral number of times the g. C. d., 40 must be an integral number of times the g. c. d. But if 40 is an exact divisor of 120 it is an exact divisor of 600, (120 x 5), and 640, (40 more than 600), and 760, (120 more than 640). We make the trial and find that it is an exact divisor of 120, and is therefore the g. c. d. of 610 and 760. From the foregoing learn that any number that is an exact divisor of two numbers is an exact divisor of their difference. 169. From the foregoing make a rule for finding the g. c. d. of two numbers and apply it to the following PROBLEMS. Find the g. c. d. 1. Of 380 and 210. 6. Of 540 and 450. 2. Of 275 and 155. 7. Of 320 and 860. 3. Of 144 and 96. 8. Of 475 and 350. 4. Of 1728 and 288. 9. Of 390 and 520. 5. Of 650 and 175. 10. Of 450 and 600. (a) Find the sum of the ten results. (P. 332.) Algebra - Equations. 170. An equation is the expression of the equality of two numbers or combinations of numbers. EQUATIONS. (3) 2x + x +4= 10 + 9 1. Every equation is made up of two members. The part of the equation which is on the left of the sign of equality is called the first member; the part on the right of the sign of equality, the second member. 2. The first member of equation No. 1 (above) is — It is made up of terms. 3. If the same number be added to each member of an equation, the equality will not be destroyed. If x=8, then x +4=8+4. If a+b=16, then a+b+c= 16+c. 4. If the same number be subtracted from each member of an equation, the equality will not be destroyed. If x=8, then x - 3= 8 – 3. If a +b=16, then a + b c=16 — c. 5. If each member of an equation be multiplied by the same number, the equality will not be destroyed. If x=8, then 4x=4 times 8, or 32. If a +b=16, then 4a +4b=4 times 16, or 64. 6. If each member of an equation be divided by the same number, the equality will not be destroyed. 8 If x= 8, then 4. 4' b 16 If a +b= 16, then + 4 4 4 or 2. a a or 4. Algebra - Equations. 7. Any term in an equation may be transposed from one member of the equation to the other; but its sign must be changed when the transposition is made. If x +5=15, then x = 15 – 5, or 10.* If x + y - .= 25, then x+y= 25 +2. 171. To find the number for which x stands, in an equation in which there is no other unknown number. EXAMPLE No. 1. Equation, x + 2x + 3x - 5=13 6x=18 X = 3 EXAMPLE No. 2. Equation, 2x + 3x + 6 = 5x – 2x + 18 2x=12 x = 6 PROBLEMS. Find the value of x. 1. x +4=12. 6. 3x + 2x – 4= x + 16. 2. r + 3x = 8. 7. 5.x - 7= 3x + 5. 3. 5x – 2= 23. 8. 7x + 2x - x= 3x + 35. 4. 3x - x= 44. 9. 5x – 4x – 3x + 6x= 44 5. 7x+x=144, 10. 6x -8 -2x = 3x + 5. (a) Find the sum of the ten results. * Observe that 5 is subtracted from each member of the equation. + Observe that 6 is added to each member of the equation. |