How many of the first 10,000 positive integers are multiples of 4 and 5, but not multiples of 6?

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Answer 1

Hi Chris,

 

Let's begin by breaking  each number down into its prime factors:

    4 = 2 x 2

    5 = 5

    6 = 2 x 3

 

Next, let's determine the Lowest Common Multiple (LCM) of the numbers 4, 5, and 6 by multiplying all common and unique prime factors of each number:

    common prime factors: 2

    unique prime factors:  2,5,3

    LCM = 2 x 2 x 5 x 3 = 60

 

Next, let's determine how many times 60 goes into 10,000 (excluding remainder):

    10,000/60 = 166 and 2/3

    Multiples of ALL 3 numbers (4,5,6) = 166 

 

Next, let's determine the Lowest Common Multiple (LCM) of the numbers 4 and 5 by multiplying all common and unique prime factors of each number:

    common prime factors:  none
    unique prime factors:  2 x 2 x 5
    LCM = 2 x 2 x 5 = 20

 

Next, let's determine how many times 20 goes into 10,000:
    10,000/20 = 500
    Multiples of BOTH numbers (4 and 5) = 500

 

Finally, let's subtract the multiples of ALL three numbers (4,5,6) from the multiples of BOTH numbers (4 and 5) to get our answer:

   Multiples of ONLY numbers 4 and 5 (excluding 6):

       500 - 166 = 334

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan.