## how to find the number of numbers divisible by both 2 and 3

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In this , I am going to discuss Worksheet on solved problem in Number System:

SOLVED PROBLEM 1:

What is the number of numbers that are divisible by both 2 and 3 from 400 to 700 ?

Solution:

In order to solve this number system question, we must apply the concept of counting rules. (rule of counting)

We will discuss the counting rules down later as and when the need arises.

In order for a number to be divisible by both 2 and 3, it is enough if the number is divisible by just 6, because the L.C.M. (least common multiple) of 2 and 3 is 6.

To understand what L.C.M. is, click the link below:

Least common multiple

For example, consider number 4: though it is divisible by 2, it is not divisible by 3.

(Can you tell me what is the test of divisibility?

4 is divisible by 2 because the remainder left after dividing 4 by 2 is 0.

Therefore, 4 is divisible by 2 as nothing is left on dividing it by 2.

4 is not divisible by 3 because the remainder left after dividing 4 by 3 is 1, i.e. not 0. That means after dividing 4 by 3, something is left, something remains.

So the test of divisibility is:  zero should be the remainder after dividing a number by another)

Next, consider 9; this is divisible by 3 but not by 2.

So, how do we find out a number which is divisible by both 2 and 3. Simple, find out the least common multiple of 2 and 3 first and then search for numbers which are divisible by this LCM of 2 and 3.

Now, what is the LCM of 2 and 3?

It is the product of 2 and 3 i.e. 2 × 3 = 6.

(to know more about LCM and the method of finding LCM of numbers which have common factors or do not, click the link below:

Least common multiple

Therefore, the multiples of 6 such as 12, 18, 24 and infinite others will be divisible by both 2 and 3.

So, to find the number of numbers divisible by both 2 and 3 from 400 to 700, inclusive, we have to find how many numbers in the given range are divisible by 6, i.e. the LCM of 2 and 3.

Now, read carefully for a wonderful method to find the number of multiples of 6 from 400 to 700, inclusive.

First find what quotient is obtained on dividing 400 by 6.

It is 66, with a remainder 4

Again, find the quotient obtained on dividing 700 by 6.

It is 116, with a remainder 4.

Now, to find with a breeze how many numbers are multiples of 6 from 400 to 700, inclusive, just subtract the quotients to find the difference of the quotients obtained above.

And the difference of the quotients is 116 – 66 = 50.

And it is 50.

50 is the number of numbers that are multiples of 6 from 400 to 700, inclusive and therefore of both 2 and 3.

What? Is that so easy? Yes.

Is it applicable for any other range say, 300 to 600 inclusive too, besides the given range.

Yes, indeed!

the method is really the same, but with some refinement to adapt to different ranges.

Oh! Is that so? Then, is the method more complex for a different range?

No, no, my dear friend!

The method is still the same and as easy as learnt by you above.

Though the discussion is lengthy and confusing, but it’s worth learning. Again, after the exact reason is explained for bringing slight alteration to adapt the method to various ranges, I will present a simple short-cut that will bring fresh lease of life to your learning.

Here goes the explanation of the modification in the method to suit for various ranges based on counting rules.

To go ahead, let us consider the range 300 to 600, inclusive.

Now, what is the number of multiples of 6 in the above range?

Following the above method, let us find out first the quotients obtained on dividing 300 and 600 by 6.

600 divided by 6 gives quotient 100 and 300 divided by 6 gives quotient 50.

Applying the above method, we find the difference in the quotients: 100 and 50 which is 50 i.e. 100 – 50 = 50.

So, is 50 the number of numbers divisible by 6 from 300 to 600, inclusive?

Not exactly.

It’s actually

50 + 1 = 51.

What? +1? But why is that?

And if yes, then why not in the range from 400 to 700?

Yes, yes my friend, its 50 + 1 and not simply the difference of the quotients 50.

Here goes the explanation:

In the range 300 to 600, inclusive, both 300 and 600 are divisible by 6.

Remember the divisibility rule?

It’s that the remainder has to be 0.

When 600 and 300 are divided by 6, then both the dividends leave remainders 0 and so they are divisible by both 6.

But the difference of the quotients 100 and 50 (obtained on dividing 600 and 300 respectively by 6) is 50.

But this difference 50 does not include one of the two between 100 (therefore the multiple 600 as a consequence) or 50 (therefore the multiple 300 as a consequence).

To appreciate this better, consider the following example:

How many integers are there from 1 to 100, inclusive.

Is it 100 or the difference 100 – 1 = 99?

It’s indeed 100.

Because, the difference:

100 and 1 = 99,  does not include both 1 and 100.

But you must count the number of integers from 1 to 100, inclusive as 100 based on the word: INCLUSIVE.

The world inclusive signifies counting integers between 1 and 100 and also the first and the last, i.e. 1 and 100.

But difference 99 (100 – 1) will not do this, i.e. will not include both 1 and 100.

One of the two between 100 and 1 gets excluded by the difference 99.

Since the exact meaning of the expression “from 1 to 100, inclusive” is inclusion of both 1 and 100, therefore a ‘1’ is added to the difference 99 to include back whichever of 1 and 100 is excluded by the difference 99.

So, the number of integers from 1 to 100, inclusive is:

Difference 99 + 1 (for inclusive counting).

Extrapolate this counting rule to the problem at hand:

Number of numbers divisible by 6 from 300 to 600, inclusive is

100 – 50 = 50 + 1 = 51.

Now, the reason of adding 1 to the difference of the quotients 50 must have dawned on you.

Because the difference of the quotients 50 will not include both 50 and 100 (and therefore respectively the dividends 300 and 600), therefore a 1 is added to this difference of the quotients.

Because both 300 and 600 are exactly divisible by 6 (as the remainders left on dividing them both by 6 is 0), so both 300 and 600 must needs be included.

But 50, the difference of the quotients 100 and 50 will exclude one of them and hence an ‘1’ is added to 50.

Therefore, the number of multiples of 6 from 300 to 600, inclusive is
50 + 1 = 51.

Then why do we not add this 1 to the difference of the quotients

116 and 66 (in the example on from 400 to 700, inclusive)

Look at the difference of the quotients 66 and 116.

66 is exactly the quotient when 396 is divided by 6, while 116 is the quotient when 696 is divided by 6

Therefore, the difference of the quotients 116 and 66 = 50 should actually be seen as the difference of the quotients obtained when 696 and 396 are divided by 6 (not 700 and 400, as they are not exactly divisible by 6, since the remainders are 4 each, when they are divided by 6 each)

Now, the difference of the quotients 116 and 66 is 50,

i.e. 116 – 66 = 50.

Now, recollect that the difference 50 will not include both 116 and 66 (just as the difference of 100 and 1 i.e. 100 – 1 = 99 does not include both 1 and 100)

So, do we add 1 to this difference of the quotients: 50.

No!

Why? See for yourself!!!

116 and 66 are quotients obtained when 696 and 396 are divided by 6 and since 396 does not lie in the range 400 to 700, so it need not be included for our purpose. But, 116 will need to be included as it is the quotient obtained on dividing 696 by 6 and since 696 lies in the range 400 to 700, so it should be included.

That is, of the two quotients 66 and 116, we need to include one and exclude the other.

Now, the difference of the quotients 116 – 66 = 50 will exactly be doing this just like 99, the difference of 100 and 1 (only integers)

Therefore, the number of multiples of 6 from 400 to 700, inclusive is 50 only.

This is how it goes.

Now, if you felt the above discussion lengthy and confusing, then go through the following short-cut to remember when ‘1’ should be added to the difference of the quotients:

Case1:

From 300 to 600, inclusive how many numbers are divisible by 6?

50 + 1

Case 2:

From 400 to 700, inclusive how many numbers are divisible by 6?

50

Case 3:

From 300 to 700, inclusive how many numbers are divisible by 6?

66 + 1

Case 4:

From 400 to 600, inclusive how many numbers are divisible by 6?

34

So, my dear friend, did you notice when all 1 was added to the difference of the quotients.

Yes, you guessed it right!

Whenever the the first number in the above four ranges is divisible by 6, then 1 must be added to the difference of the quotients.

And this method will work to find how many numbers are divisible by two factors in various ranges such as 400 to 700 and so on.