MHB Count of Multiples of 2 or 7 in 999

[sabsac]

Find the count of multiples of 2;or 7, for all natural numbers less than or equal to 999.

 

[HallsofIvy]

There are integer part of 999/2= 499 even numbers les than or equal to 999. There are integer part of 999/7= 142 multiples of 7 less than or equal to 999. But 499+ 142= [FONT=Verdana,Arial,Tahoma,Calibri,Geneva,sans-serif]1141 is too larger because it counts multiples of 14 twice. We need to subtract integer part of 999/14= 71 to account for that.

 

[sabsac]

the answer provided is 570 which is the difference between 641 and 71. that difference between the total count of multiples of 2 and 7.

 

[Opalg]

There are integer part of 999/2= 499 even numbers les than or equal to 999. There are integer part of 999/7= 142 multiples of 7 less than or equal to 999. But 499+ 142= 1141 is too larger because it counts multiples of 14 twice. We need to subtract integer part of 999/14= 71 to account for that.

Mathematicians often have trouble with simple arithmetic!

 

[sabsac]

so can you once again explain how we arrived at the value 71 and why we had to subtract it from 641?

 

[skeeter]

multiples of 2 less than 999 ...

2(1), 2(2), 2(3), ... , 2(499)

multiples of 7 less than 999 ...

7(1), 7(2), 7(3), ... , 7(142)

multiples of 14 less than 999 which are common to both lists above ...

14(1), 14(2), 14(3), ... , 14(71)
number of multiples of 2 or 7 that are less than 999 =

(multiples of 2)+(multiples of 7)-(number of values that are multiples of both 2 and 7) =

499+142-71

 

[sabsac]

Thanks for clarifying.