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Properties of Numbers.

36 = 2x2x3x3. 120 = 2x2x2x3x5. The 1. c. m. of 36 and 120 must contain three 2's, two 3's, and one 5. 2 x 2 X 2 x 3x3x5= 360, 1. c. m. of 36 and 120.

6. To find the 1. c. m. of two or more numbers: Resolve each number into its prime factors. Take as factors of the 1. c. To. the greatest number of 2's, 3's, 5's, 7's, etc., found in any one of the numbers.

Example.
Find the 1. c. m. of 24, 35, 36, and 50.

Operation.
24 = 2x2x2x3.

35 = 5 x 7.

36 = 2x2x3x3.
50 = 2 x 5 x 5.

2x2x2x3x3x5x5x7 = 12600, 1. c. m.

Explanation.

24 has the greatest number of 2's as factors.

36 has the greatest number of 3's as factors.

50 has the greatest number of 5's as factors.

35 is the only number in which the factor 7 occurs. There must be as many 2's among the factors of the 1. c. m. as there are 2's among the factors of 24; as many 3's as there are 3's among the factors of 36; as many 5's as there are 5's among the factors of 50; as many 7's as there are 7's among the factors of 35; that is, three 2's, two 3's, two 5's, and one 7.

Find the 1. c. m.:

7. Of 48 and 60. 11. Of 20, 30, and 40.

8. Of 60 and 75. 12. Of 40, 50, and 60.

9. Of 50 and 60. 13. Of 24, 48, and 36. 10. Of 30 and 40. 14. Of 25, 35, and 40. Algebra—Parentheses.

154. When an expression consisting of two or more terms is to be treated as a whole, it may be enclosed in a parenthesis.

12+ (5+ 3) =? | 7a + (3a + 2a) =?

12 + 5 + 3 =? | la + 3a + 2a =?

Observe that removing the parenthesis makes no change in the results.

12 - (5 + 3) =? I 7a - (3a + 2a; =?

12 - 5 - 3 =? I 7a-- 3a - 2a =?

Observe the change in signs made necessary by the removal of the parenthesis.

| 12-(5-3) =? | 7a -(3a- 2a) =?

\ 12-5 + 3 =? (7a-3a + 2a =?

Observe the change in signs made necessary by the removal of the parenthesis.

A careful study and comparison of the foregoing problems will make the reasons for the following apparent:

I. If an expression within a parenthesis is preceded by the plus sign, the parenthesis may be removed without making any changes in the signs of the terms.

II. If an expression within a parenthesis is preceded by a minus sign, the parenthesis may be removed; but the sign of each term in the parenthesis must be changed; the sign +"to -, and the sign to +.

155. Remove the parenthesis, change the signs if necessary, and combine the terms:

1. 15 - (6 + 4) = 5. 156 - (126 - 46) =

2. 18 + (4 - 3) = 6. 18c + (9c - 3c) =

3. 27-(8 + 3)= 7. 2id-(5d + 3d) =

4. 45+(12-3)= 8. 36jj-(5* + 4a;) =

Algebra—Parentheses.

156. Multiplying An Expression Enclosed In A
Parenthesis.

1. 6(7 + 4) =• ?* 6(7a + 46) =?

6(a + 6) =? Ans. 6a + 66.

a(b -j- c) =? Ans. a6 + acObserve that in multiplying the sum of two numbers by a third number, the sum may be found and multiplied; or each number may be multiplied and the sum of the products found.

In the last three examples given above, substitute 5 for a, 3 for b, and 2 for c; then perform again the operations indicated, and compare the results with those obtained when the letters were employed.

2. 6(7 - 4) =? 6(7a - 46) =?

6{a b) 1 Ans. 6a 6b.

a(b c) =? Ans. ab ac. Observe that in multiplying the difference of two numbers by a third number, the difference may be found and multiplied; or each number may be multiplied and the difference of the products found.

In the last three problems given above, substitute 5 for a, 3 for b, and 2 for c; then perform again the operations indicated, and compare the results with those obtained when the letters were employed.

lBY Problems. If a = 5, b = 3, and c = 2, find the value of the following:

1. 3(a + b) - 2(6 + c).

2. i(a + 26) - 3(6 - c).

3. 2(2a - 6) + 2(26 - c).

* This means, that the sum of 7 and 4 is to be multiplied by six; or that the sum of six 7's and six 4's is to be found.

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Right Triangle.

1. The sum of the angles of any triangle is equal to

right angles or degrees.

2. In a right triangle there is one right angle. The other two angles are together equal to .

3. In a certain right triangle one of the angles is an angle of 40°. How many degrees in each of the other two angles? Draw such a triangle.

4. Convince yourself by drawings and measurements that every equilateral triangle is equiangular.

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5. Note that in every equiangular triangle each angle is one third of 2 right angles. So each angle is an angle of ——. degrees.

6. If any one of the angles of a triangle is greater or less than 60, can the triangle be equiangular? a Can it be equilateral?

7. If angle a of an isosceles triangle measures 50°, how many degrees in angle bl In angle c?

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159. Miscellaneous Review.

1. I am thinking of a right triangle one of whose angles measures 32°. Give the measurements of the other two angles. Draw such a triangle.

2. I am thinking of an isosceles triangle; the sum of its two equal angles is 100°. Give the measurement of its third angle. Draw such a triangle.

3. Let a equal the number of degrees in one angle of a triangle and b equal the number of degrees in another angle of the same triangle; then the number of degrees in the third angle is 180° — (a + b). If a equals 30, and b equals 45, how many degrees in the third angle?

4. Name three common multiples of 16 and 12.

5. Name the least common multiple of 16 and 12.

6. Find the sum of all the prime numbers from 101 to 127 inclusive.

7. Find the prime factors of 836.

8. With the prime factors of 836 in mind or represented on the blackboard, tell the following:

(a) How many times is 19 contained in 836?

(b) How many times is 11 x 19 contained in 836?

(c) How many times is 19 x 11 X 2 contained in 836?

160. Problems. Find the l. c. m.

1. Of 18 and 20. 6. Of 36, 72, and 24.

2. Of 13 and 11. 7. Of 45, 81, and 27.

3. Of 24 and 32. 8. Of 33, 55, and 88.

4. Of 16 and 38. 9. Of 45, 65, and 85.

5. Of 46 and 86. 10. Of 3, 5, 7, and 11. (a) Find the sum of the ten results.

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