Hints for solving the problems - Apply 1.2

GCF – Greatest Common Factor

LCM –Lowest Common Multiple

LCD – Lowest Common Denominator (Same as LCM except these problems

   are fractions)

The GCF is the largest number that can be divided into the given numbers evenly.

The LCM is the smallest number that the given numbers will divide into evenly.

Problem No. 2 - Find the GCF and LCM of 10 and 36.

First change each number to its prime factored form.

Prime numbers are numbers that can be divided by only one and itself evenly. Such as: 2, 3, 5, 7, 11, 13, 17, 19, & etc.

        10 = 2 · 5

The GCF is the factors that are common to both numbers. Also use the smallest number times the common factors are used in any one number.

For this group 2 is the only factor that is common and the smallest time it is used is once.

           So the  GCF  is  2.

The LCM is all the different factors shown in both numbers and the largest times it is used in any one number.

The different factors are: 2, 3, and 5 – but in the second number the 2 is shown twice and the 3 is shown twice.

           So the LCM is:   2 · 2 · 3 · 3 · 5   =   180.

Problem No. 14 – Find the GCF and LCM of 32 and 48.

Prime factorization of   32   is   2 · 2 · 2 · 2 · 2

Prime factorization of   48   is   2 · 2 · 2 · 2 · 3

                      The GCF  is  2 · 2 · 2 · 2   =   16

The 2 is the only common factor for both numbers and the smallest number times it is used is 4.

                       The LCM  is   2 · 2 · 2 · 2 · 2 · 3   =   96

The different factors are 2 and 3 and the largest number of times the 2 is used is 5 and the largest number of times the 3 is used is 1.

Problem No. 26  –  Find the GCF and LCM of 48, 72, and 120

Prime factorization of  48  is   2 · 2 · 2 · 2 · 3

Prime factorization of  72  is    2 · 2 · 2 · 3 · 3

Prime factorization of 120  is   2 · 2 · 2 · 3 · 5

                     The GCF  is  2 · 2 · 2 · 3   =   24

                      The LCM  is  2 · 2 · 2 · 2 · 3 · 3 · 5   =   720

Problem No. 30  –  Write in lowest terms:   

This problem is reducing fractions. You can find numbers that will divide into the numerator and denominator and reduce that way or use the prime factorization method.

        The answer is  .

Problem No. 34  –  Find:   

Multiplication of fractions – Reduce through cancellation (reducing by division) and multiply the numerators together and the denominators together.

       The answer is  .

Problem No. 40  –  Find:    

Division of fractions – Invert the fraction on the right of the division sign and perform the rules of multiplications of fractions.

          The answer is  

Problem No. 42  –  Find:   

To add or subtract fractions there must be a common denominator (LCD). If it is already given, just add the numerators and keep the same denominator and make sure you reduce your answer to lowest terms.

             The answer is 1.

Problem No. 48  –  Find:   

The denominators are different, so an LCD must be found.

        30  =  2 · 3 · 5

        35  =  5 · 7

     The LCD  is  2 · 3· 5· 7  =  210

All fractions must be changed to show this LCD.

             The answer is  

Remember raise all fractions to show the LCD by multiplying both the numerator and denominator by the same number. That number is what makes the denominator show the LCD.

Problem No. 56  –  Find:  

Subtraction is the same procedure as addition, but you subtract the numerator after the fractions have been raised to show the LCD.

              16   =  2 · 2 · 2 · 2

              24   =   2 · 2 · 2 · 3

      The LCD  is  2 · 2 · 2 · 2 · 3  =  48

               The answer is