Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

SIGNS OF OPERATION

A parenthesis () or a vinculum groups together several numbers and shows that the operations within the groups are to be performed first; thus, 6-(3+2)=6-5=1; (5+3) × 2=8 x 2 = 16; 5+ (3 × 2)=5+6=11; 32 ÷ 4 + 3 =8+3=11; 4 x 2-3=8-3=5.

When no parenthesis or vinculum is used, the signs × and indicate operations that are to be performed before those indicated by either + or - ; thus, 4+8 × 3 = 4+ (8 × 3), or 28; 5+12÷6 = 5+ (12 ÷ 6), or 7.

In an expression like 12÷6 × 2, mathematicians are not agreed as to which sign shall be used first. To avoid ambiguity, the parenthesis should be used in such expressions. Thus, (126) × 2=4; but 12 ÷ (6 × 2) = 1.

[blocks in formation]

10. 7 x 5+ 4+8×6 ÷ 2 − 3 × 4.

11. (6 + 2 × 3) ÷ + + (3 × 6) ÷ 2 + 2 × (3 + 5 − 2).

12. 36 − 6 × 4 + 2 × 6 + (40 + 5) ÷ 9 + 3 × 6.

13.

10+205x3+6×2÷3+5 × 6.

14. 3 × (4 +5 −2) + + + 5 × (4 × 5 ÷ 2) + 5.

15. 3 × (68) + 7 × (8 ÷ 2) − 3 × (6 ÷ 3) + 15 −7.

16. 175-8 x (1910) - 25+5+6x7-9+ 3.

FACTORS AND DIVISORS

1. What two numbers will give 6 as a product? 8 as a product? 10 as a product?

2. What are 2 and 3 in relation to 6? 4 and 2 in relation to 8? 5 and 2 in relation to 10?

An integer or an integral number is a whole number.

The factors of a number are the integers whose product is the number; thus, 5 and 2 are factors of 10.

3. Name two factors that produce 24, 32, 40, 56, 49, 72, 96. A factor of a number is an exact divisor of the number: ; that is, it is contained in the product an integral number of times.

4. Name the exact divisors of 54, 81, 48, 36, 66, 64, 63. 2 x2 = 4 = 9

5. Observe the two equal factors that pro- 3×3 duce 4; 9; 16.

4 x 4 = 16

2 × 2 × 2

6. Observe the three equal factors that pro- 3x3x3=27 duce 8; 27; 64.

4 x 4 x 4 = 64

Instead of repeating a factor, a small figure called an exponent may be written to the right and a little above the number to show how often it is used as a factor; thus, 33 = 3 x 3 x 3 = 27; 24 = 2 × 2 × 2 × 2 = 16.

7. What number will divide 9 and 10? 21 and 25?

Numbers are prime to each other when they have no common factor; thus, 9 and 10 are prime to each other.

Even numbers are numbers that contain the factor 2.

Odd numbers are numbers that do not contain the factor 2.

8. What are the factors of 7? of 11? Observe that 7 and 11 have no exact divisors except themselves and one.

A prime number is one that has no exact divisor except itself and one; thus, 5, 2, and 3 are prime numbers.

9. Name all the prime numbers to 31.

10. What are the factors of 15? Observe that 15 can be divided by 3 and 5. It is composed of other factors than itself and one.

A composite number is one that has other exact divisors than itself and one; thus, 6 and 10 are composite numbers. 11. Name all the composite numbers to 50.

TESTS OF DIVISIBILITY

1. Divide 12, 24, 26, 38, and 50 each by 2. What is the ones' figure in each of the dividends? Divide other numbers ending in 2, 4, 6, 8, or 0 by 2.

A number is divisible by 2, if the ones' figure is 2, 4, 6, 8, or 0.

2. Divide 15, 25, 40, 125, 150 each by 5. What is the ones' figure in each dividend? Divide other numbers ending in 5 or 0 by 5.

A number is divisible by 5, if its ones' figure is 5 or 0.

Notice

3. Divide 36, 69, 48, 72, 162, 369 each by 3. that the sum of the digits (that is, of the figures) in each number is divisible by 3. Divide by 3, other numbers the sum of whose digits is divisible by 3.

A number is divisible by 3, if the sum of its digits is divisible by 3.

4. Divide 18, 27, 279, 819, 639 each by 9. Notice that the sum of the digits in each dividend is divisible by 9. Divide by 9, other numbers the sum of whose digits is divisible by 9.

A number is divisible by 9, if the sum of its digits is divisible by 9.

5. Select the numbers that are divisible by 2; by 3; by 5; by 9.

3672

94

86 96 123 918 515
72 321 819 450

1909

FACTORING

1. Give the two factors that produce 15.

2. If one of them is given, how may the other be found? To separate a number into two factors, take any exact divisor for one factor and the quotient of the number by this factor for the other.

Factoring is the process of separating a number into its factors.

A prime factor is a prime number used as a factor; thus, 3 and 5 are the prime factors of 15.

Written Work

1. Find the prime factors of 126.

23

2 | 126
63

3

21
7

Divide by the least prime factor; divide the quotient by the next smallest prime factor, etc., until the last quotient is a prime number. The divisors and

Test: 2 × 3 × 3 × 7=126 the last quotient are the prime factors;

Or,

2 x 32 x 7 = 126

thus, 2, 3, 3, and 7 are the prime factors of 126.

[blocks in formation]
[blocks in formation]

1. Name a number that will exactly divide both 16 and 24; 15 and 25; 14 and 27.

A common divisor of two or more numbers is a number that exactly divides each of them; thus, 4 is a common divisor of 16 and 24.

2. Is 4 the greatest number that will exactly divide 16 and 24? What is the greatest number that will exactly divide 16 and 24?

The greatest common divisor (g. c. d.) of two or more numbers is the greatest number that exactly divides each of them; thus, 9 is the g. c. d. of 27 and 36.

3. Name the g. c. d. of 24 and 36; of 32 and 40.

Written Work

1. Find the greatest common divisor of 56, 98, 154.

256 98 154

7 28 49 77

4

7

11

As the g.c.d. of two or more numbers is the product of all their common prime factors, divide the numbers by their common prime factors. In the same way divide the quotients until g.c.d=2x7, or 14. they are prime to each other. The divisors 2 and 7 are all the common prime factors of the numbers. Hence the g.c.d. of 56, 98, and 154 is 2 x 7, or 14.

« ΠροηγούμενηΣυνέχεια »