d.: Divisibility of Numbers. 6. Find the g. c. d. of 640 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; so is the number 640. We make an incom120)640(5 600 plete division of 760 by 640 and have as a remainder the number 120. Since 640 and 40)12013 760 are each an integral number of times 120 the g. c. d., their difference, 120, must be an integral number of times the g. c. 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 divisor of 640, it is also an exact divisor of 760, for it will be contained one more time in 760 than in 640. 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 640 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 240. 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. Algebra-Equations. 170. An equation is the expression of the equality of two numbers or combinations of numbers. If x = EQUATIONS. (3) 2 x + 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. 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 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 4 x 4 times 8, or 32. If a + b 16, then 4 a + 4b 4 times 16, or 64. 6. If each member of an equation be divided by the sanie number, the equality will not be destroyed. 8 16 8 – 3. = x or 2. 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.* 2 = 25, then x + y 171. To find the number for which x stands, in an equation in which there is no other unknown number. 18 – 6. 25 +2. If x + y EXAMPLE No. 1. 18 x 3 6 x = EXAMPLE No. 2. 12 6 2 x = = 12 PROBLEMS. Find the value of x. 1. x + 4 6. 3 x + 2 x – 4 = x + 16 2. x + 3 x = 8 7. 5 x - 7 = 3 x + 5 3. 5 2 23 8. 7 x + 2 x 3 x + 35 4. 3 x x = 44 х X = 9. 5 x 3 x + 6 x 44 5. 7 x + x = 144. 10. 6 x - 8 - 2 x = 3 x + 5 (a) Find the sum of the ten results. - 4x * Observe that 5 is subtracted from each member of the equation. t Observe that 6 is added to each member of the equation. 1. Two of the sides of a trapezoid are parallel and two are not parallel. In the trapezoid represented above the side ac is parallel to the side 2. No two of the bounding lines of a trapezium are parallel. 3. In the trapezoid represented above no one of the angles is a right angle. Name the angles that are greater than right angles; the angles that are less than right angles. 4. Draw a trapezoid two of whose angles are right angles. 5. Can you draw a trapezoid having one and only one right angle? 6. Draw a trapezium one of whose angles is a right angle. 7. Can you draw a trapezium having more than one right angle? 8. Every quadrilateral (trapezium, 'trapezoid, or parallelogram) may be divided into two triangles. Remember that the sum of the angles of two triangles is equal to four right angles. Observe that the sum of the angles of the two triangles is equal to the sum of the angles of the quadrilateral. So the sum of the angles of a quadı ilateral is equal to four right angles. 173. MISCELLANEOUS REVIEW. 1. If two of the angles of a trapezoid are right angles and the third is an angle of 60°, how many degrees in the fourth angle? Draw such a trapezoid. * 2. If the sum of three of the angles of a trapezium is 298°, how many degrees in the fourth angle? Draw such a trapezium. 3. If one of the angles of a triangle is an angle of 80°, and the other two angles are equal, how many degrees in each of the other angles ? Draw the figure. * 4. If one of the angles of a quadrilateral is a right angle, and the other three angles are equal, what kind of a quadrilateral is the figure? 5. One of the angles of a quadrilateral is a degrees; another is b degrees; the third is c degrees. How many degrees in the fourth angle 6. The smallest angle of a triangle is x degrees; another angle is 2 x degrees, and the third is 3 x degrees: Then x + 2 x + 3 x = 180. Find the value of x; of 2 x; of 3 x. 7. 643,265,245,350. Without performing the division tell whether this number is exactly divisible by 9; by 5; by 10; by 25; by 50; by 123; by 18; by 6; by 15; by 30; by 90; by 163.+ 8. A number is made up of the following prime factors: 2, 2, 3, 3, 5, 7, 11. Is the number exactly divisible by 18? by 26 ? by 35 ? by 77 ? by 21 ? by 30? by 45 ? by 8 ? * It is not expected that this drawing will be accurate in its angular measurements-simply an approximation to accuracy, to aid the pupil in recognizing the comparative size of angles. + A careful study of pages 71-75 inclusive will enable the pupil to make the statements called for, with little hesitation. See problem 3, page 70. |