Boolean Algebra/How many numbers between 1 and 1000

[johndoe88]

Hey Guys,

I have been having difficulty with converting problems into a logical boolean sentence in order to solve them. In the problem below I am not so much concerned with finding the answer but writing out the sentence in order to find the solution. Maybe if there is a guide/book/etc. that can help me out with this subject.

For example: How many numbers between 1 and 1000 are divisible by 2 and 3?

Let A=total amount of numbers divisible by 2 between 1 and 1000
B=total amount of numbers divisible by 3 between 1 and 1000
T=total amount of numbers between 1 and 1000

So AnB=A+B-AuB

1) How many numbers between 1 and 1000 are not divisible by 2 and 3?
~(AnB)=~Au~B (Demorgan's Law)=~A+~B-(~An~B)
could we also write ~(AnB)=T-(AnB)
OR is this question asking ~An~B which does not equal ~(AnB)
~An~B

2) How many numbers between 1 and 1000 are divisible by 2 but not divisible by 3?
An~B=A+B-(Au~B)

3) How many numbers between 1 and 1000 are divisible by either 2 or 3?
AuB=A+B-AnB

4) How many numbers between 1 and 1000 are divisible by neither 2 or 3?
~(AuB)=T-(AuB)
where AuB=A+B-AnB

[sparkles3144]

How many numbers between 1 and 1000 are divisible by 2 and 3?

How many numbers between 1 and 1000 are divisible by 6?

6 is the smallest number that is divisible by both 2 and B.

1000/6 = 166.67

6*1 = 6
6*166 = 996

So total of 166 numbers between 1 and 1000 are divisible by 2 and 3?

[[email protected]]

Hi johndoe88,

Sparkles3144 shows a nice, easy way to solve this problem.

GMAT questions usually can be solved in several different ways. One of the skills that Business Schools are looking for is your ability to be correct AND flexible. So, be careful about trying to solve a problem in a complicated or overly-technical way - there might be a much faster, easier way to answer the question. By training yourself in these other tactics, you'll move faster and likely have to do less work. This will affect your mindset, endurance, pacing and overall score in positive ways.

GMAT assassins aren't born, they're made,
Rich

[johndoe88]

Rich, I think there was a misunderstanding about my question. I am not necessarily looking for the solution as the sentence used to calculate the solution.

How about:

2) How many numbers between 1 and 1000 are divisible by 2 but not divisible by 3?

[johndoe88]

Rich, I think there was a misunderstanding about my question. I am not necessarily looking for the solution as the sentence used to calculate the solution.

How about:

2) How many numbers between 1 and 1000 are divisible by 2 but not divisible by 3?

[[email protected]]

Hi johndoe88,

In these situations, you have to define what you're after and then go after it.

In your example, we want to know how many numbers between 1 and 1000 (inclusive) are multiples of 2 but NOT multiples of 3.

So, we need to total up the multiples of 2 and then remove the numbers from THAT group that are multiples of 3....

From 1 to 1,000 there are 500 multiples of 2 (2, 4, 6, 8, 10, 12, etc.)

You'll notice that every THIRD number is a multiple of 3 (6, 12, etc.), so we have to remove those from the 500 numbers we have.

500/3 = 166r2 numbers we must remove. You'll notice that the remainder accounts for the terms that come after the "last" multiple of 3.

In this case, the last few numbers would be 1000, 998, 996 -----996 is a multiple of 3 and is removed; 998 and 1000 are NOT multiples of 3, so we have to include them in the total (those 2 numbers are represented by the r2 in the calculation).

In the end, we have 500 - 166 numbers = 334 numbers

GMAT assassins aren't born, they're made,
Rich

[johndoe88]

Rich,

I understand your logic. Now all I am asking is for the Boolean formula that your using.

2) How many numbers between 1 and 1000 are divisible by 2 but not divisible by 3?

Let A=total amount of numbers divisible by 2 between 1 and 1000
B=total amount of numbers divisible by 3 between 1 and 1000
T=total amount of numbers between 1 and 1000

An~B=A-AnB
In this case AnB=Divisible by 2 and divisible by 3=amount of numbers divisible by 6
Is this formula correct? Can I it from now on? How about ~AnB

Though I am also curious: You found that the number of even numbers between 1-1000 is 500. To find the number of multiple of 6 in the sequence of even numbers you divided 500(number of even multiples by 3). What is the logic behind the formula?

If you wanted to find the number of 7 multiples between 1 and 100. You can
(last multiple - first multiple )/7+1=(98-7)/7+1=14
The shortcut you used 100/7=14.something=14
I reason why the shortcut formula works is because the interval of numbers starts at 1. If I had asked find how many multiples of 7 between 50 and 100, you couldn't just divide 100/7 and say 14. But the shortcut formula worked for the series (2,4,6,8....996, 998).

[johndoe88]

Anyone?? Greatly appreciate the help.

[sanju09]

Hey Guys,

I have been having difficulty with converting problems into a logical boolean sentence in order to solve them. In the problem below I am not so much concerned with finding the answer but writing out the sentence in order to find the solution. Maybe if there is a guide/book/etc. that can help me out with this subject.

For example: How many numbers between 1 and 1000 are divisible by 2 and 3?

Let A=total amount of numbers divisible by 2 between 1 and 1000
B=total amount of numbers divisible by 3 between 1 and 1000
T=total amount of numbers between 1 and 1000

So AnB=A+B-AuB

1) How many numbers between 1 and 1000 are not divisible by 2 and 3?
~(AnB)=~Au~B (Demorgan's Law)=~A+~B-(~An~B)
could we also write ~(AnB)=T-(AnB)
OR is this question asking ~An~B which does not equal ~(AnB)
~An~B

2) How many numbers between 1 and 1000 are divisible by 2 but not divisible by 3?
An~B=A+B-(Au~B)

3) How many numbers between 1 and 1000 are divisible by either 2 or 3?
AuB=A+B-AnB

4) How many numbers between 1 and 1000 are divisible by neither 2 or 3?
~(AuB)=T-(AuB)
where AuB=A+B-AnB

Is it a repeat post?

https://www.beatthegmat.com/boolean-alge ... tml#688115