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

[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

In Group Questions on two sets, all we need to remember is that in order to find the number of elements in the required set of numbers out from two different sets,

1. ADD the number of elements in each set,
2. SUBTRACT the number of common elements in the two sets from the result of step 1,
3. ADD to the result of step 2 the number of elements that do not belong to either set.

And we are done!

Or we can simply remember this in short, like

REQUIRED = ONE + TWO - BOTH + NEITHER

Back to the questions now...

Between 1 and 1000 means, 1 and 1000 are not included. In the opening case, in other words, we are looking for the number of multiples of 6 between 1 and 1000, and the answer is simple to find, smallest is 6, greatest is 996, then the easy steps to remember, difference is 990, divided by 6 is 165, add 1 gives 166, the answer.

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

Things are more simple now. There are a total of (1000 - 1) - 1 = 998 positive integers between 1 and 1000, and out of these 998, exactly 166 are divisible by 6; hence 998 - 166 = 832 numbers between 1 and 1000 are not divisible by 2 and 3. The answer.

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

So we now have two sets of numbers between 1 and 1000 to deal with, one set incudes divisible by 2, and the other set includes not divisible by 3. Quick count for the first set, {(998 - 2)/2} + 1 = 499, for the second set it must be 998 minus divisible by 3, which is same as 998 - [{(999 - 3)/3} + 1] = 998 - 333 = 665. Out of these 665, only (665 - 1)/2 = 332 numbers between 1 and 1000 that are divisible by 2 but not divisible by 3. The answer.

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

We have 499 numbers divisible by 2, 333 numbers divisible by 3, and 166 numbers divisible by both. Here, we won't consider the numbers that are divisible by neither, then applying: Required = I + II - Both, we get 499 + 333 - 166 = 666, as answer.

4. How many numbers between 1 and 1000 are divisible by neither 2 or 3?

The leftover numbers in the list, apart from those divisible by either 2 or 3, is the number to answer this question. Just do 998 - 666 = 332, and answer.

GMAC never expects a test taker know the terms and concepts pertaining to Boolean Algebra or the De-Morgan's Law. Although it expects us to deduce logically as we do in the Boolean Algebra or in the De-Morgan's Law.