ALGEBRA - solved problems
  Natural numbers and integers, rules and properties
Operations on integers
11.    Add integers with same signs.
Solution:   To add integers having the same sign, keep the same sign and add the absolute value of each number.
12.    Add integers with different signs.
Solution:   To add integers with different signs, keep the sign of the number with larger absolute value and subtract smaller absolute value from the larger.
13.    Subtract given integers:
Solution:   Subtract an integer by adding its opposite,
                          a)   5 -  7 =  5 + (- 7) = - 2,             b)   - 1 -  8 =  - 1 - ( + 8) = - 1+ (- 8) = - 9, 
                          c)   4 - ( - 3) =  4 + ( + 3) = 7,          d)   - 5 -  (- 9) =  - 5 + (+ 9) = - 5+ 9 = 4.
14.    The use of parentheses: 
Solution:   Evaluate given expressions,
                          a)   (- 5 + 3) + 6 = - 5 + 3 + 6 = 4,             b)   - 7 + (- 3 + 8) = - 7 - 3 + 8 = - 2, 
                          c)   6 - ( 5 - 12)  = 6 -  5 + 12 = 13,            d)   - (- 2 + 7) + 3 = 2 - 7 + 3 = - 2.
Multiplication and division of integers
15.    Multiply given integers: 
Solution:            a)   7 · 0 = 0,                       b)   (- 1) · 18 = - 18, 
                          c)   (- 3) · (- 4) = 12,             d)   3 · (- 4) = -12.
16.    Multiply given integers: 
Solution:            a)   (- 3 + 5) · 6 = (- 3) · 6 + 5 · 6 = - 18 + 30 = 12,
                          b)   ( 7 -  4) · (- 3) = 7 · (- 3) - 4 · (- 3) = - 21+ 12 = - 9,
                          c)   (- 5 ) · 3 + 2 · 3 =  (- 5 + 2) · 3 = (- 3) · 3 = - 9,
                          d)   9 · (- 6) - 2 · (- 6) = (9 - 2) · (- 6) = 7 · (- 6) = - 42.
17.    Solve:
Solution:          a)   (- 7 + 4) · (- 2) = (- 7) · (- 2) + 4 · (- 2) = 14 -  8 = 6       or        (- 3) · (- 2) = 6,
                         b)   - 9 · a + 5 · a = (- 9 + 5) · a = - 4a,        c)   3 · (1 - a) + 5 = 3 - 3 · a + 5 = 8 - 3a.
18.    Divide given integers: 
Solution:           a)   0 ¸ 5 = 0,                                   b)   - 3 ¸ 1 =  - 3, 
                          c)   - 28 ¸ (- 4) = 28 ¸ 4 = 7,            d)  45 ¸ (- 15) = - 45 ¸ 15 = - 3.
If both the dividend and divisor signs are the same the quotient will be positive, if they are different, the quotient will be negative.
19.    Divide given integers: 
Solution:            a)   (36 ¸ 6) ¸2 = 6 ¸ 2 = 3,      while      b)   36 ¸(6 ¸ 2)= 36 ¸ 3 = 12, 
                          c)   18 ¸ 6 ¸ 3 = 3 ¸ 3 = 1,       while      d)  18 ¸(6 ¸ 3)= 18 ¸ 2 = 9.
Order of operations - parenthesis as grouping symbols
20.    Solve:
Solution:          a)   - 8 -  3 · (- 4) + 15 ¸ (- 5) = - 8 - (- 12) + (- 3) = - 8 + 12 - 3 = 1.
When expressions have more than one operation, we have to follow rules for the order of operations.
First do all multiplication and division operations working from left to right. Next do all addition and subtraction.
                        b)   - 3 + 2 · [- 2 - 3 · (- 4 ¸ 2 + 1)] = - 3 + 2 · [- 2 - 3 · (- 2 + 1)]
                                   = - 3 + 2 · [- 2 - 3 · (- 1)] = - 3 + 2 · [- 2 + 3] = - 3 + 2 = - 1.
We often use grouping symbols, like parentheses, to help us organize complicated expressions into simpler ones. 
Do operations in parentheses and other grouping symbols first. If there are grouping symbols  within other grouping symbols do the innermost first.
The integer prime-factorization
21.    Find the prime factors of  350.
Solution:    The sequence  of prime numbers begins  2, 3, 5, 7, 11, 13, 17, 19, 23, ...
Every positive integer greater than 1 can be factored as a product of prime numbers.
Given a number n divide by the first prime from the list of primes, if it does not divide cleanly, divide n by the next prime, and so on.
Thus, the prime factors of  350 = 2 · 5 · 5 · 7.
350 | 2
175 | 5
35 | 5
7 | 7
  1 | 
The greatest common divisor (GCD)
22.    Find the greatest common divisor of  72 and 90.
Solution:    The greatest common divisor of two integers is the largest integer that divides both numbers.
The greatest common divisor can be computed by determining the prime factors of the two numbers and comparing the factors,
72 = 2 · 2 · 2 · 3 · 3,      90 = 2 · 3 · 3 · 5,    =>     GCD(72, 90) = 2 · 3 · 3 = 18.
The least common multiple (LCM)
23.     Find the least common multiple or LCM of,  24, 54 and 60.
Solution:    A common multiple is a number that is a multiple of two or more numbers. 
The least common multiple of two integers a and b is the smallest positive integer that is a multiple of both a and b. To find the least common multiple of two numbers:
 - first list the prime factors of each number, then
- multiply each factor the greatest number of times it occurs in either number.
If the same factor occurs more than once in both numbers, multiply the factor the greatest number of times it occurs.
Thus, the prime factors:    24 = 2 · 2 · 2 · 3,    54 = 2 · 3 · 3 · 3,    60 = 2 · 2 · 3 · 5,
=>       LCM(24, 54, 60) = 2 · 2 · 2 · 3 · 3 · 3 · 5 = 1080.
  Operations on real numbers
24.    Add or subtract given real numbers:
Solution:       a)   −7 − 4 = − 7 + (− 4) = − 11,                b )   + 3 − 9 = 3 + (− 9) = − 6,
                     c)     5 − (− 6) = 5 + (+ 6) = 11,                 d )    −3 − (− 9) =  −3 + 9 = 6,
                     e)     0.5 − 1.5 = 0.5 + (− 1.5) = − 1,           f )    −1.5 − 0.25 = −1.5 + (− 0.25) = −1.75.
25.    Multiply or divide given real numbers:
Solution:       a)   −7 · (− 4) = 28,            b)   3 · (− 9) = − 27,             c)   − 6 · 0.1 = − 0.6,
                     d)   6 ¸ (− 0.1) = 6 · (− 10)  = − 60,        e)   − 8 ¸ (− 0.125) = − 8 · (− 1000/125) = 64
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