Question

. How many numbers up to 700 are divisible by 4 or 5?

Answer 1

Given :- How many natural numbers up to 700 are divisible by 4 or 5 ?

Solution :-

The first number divisible by 5 is 5 and second number is 10 and so on. just keep adding 5 till you reach the last number which is 700 .

So, the AP series is :-

• 5, 10, 15, 20________ ,700. { upto means we have to take 700 also. }

Now to find the total number of numbers put it in the formula of n terms of AP .

• Tn = a + (n - 1)d = 700
• a = first term = 5
• d = common difference = 5 .

→ 700 = 5 + (n - 1)*5

→ 700 = 5 + 5n - 5

→ 5n = 700

→ n = 140

Similarly, for number divisible by 4 we get the AP series is :-

• 4, 8 , 16 , ___________ 700.

again, putting in nth term formula we get,

→ 700 = 4 + (n - 1)4

→ 700 = 4 + 4n - 4

→ 4n = 700

→ n = 175 .

Now we must take care of repetition that is those numbers which is divided by 4 and 5 both .

Those numbers are:-

• 20, 40, 60, _____________ 700 . { LCM of both. }

So, again using AP nth term formula,

→ 700 = 20 + (n - 1)20

→ 700 = 20 + 20n - 20

→ 20n = 700

→ n = 35 .

Therefore,

→ Total number of terms divisible by 4 or 5 is = (140 + 175) - 35 = 280 (Ans.)

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Answer 2

SOLUTION :

TO DETERMINE

The total number of numbers up to 700 are divisible by 4 or 5

EVALUATION

Let

A = The set of numbers divisible by 4 upto 700

B = The set of numbers divisible by 5 upto 700

We know that

[ x ] = The greatest integer but not greater than x

$\displaystyle \sf{} n(A) = \bigg[ \frac{700}{4} \bigg] = 175$

$\displaystyle \sf{} n(B) = \bigg[ \frac{700}{5} \bigg] = 140$

Now we know that if a number is divisible by both 4 and 5 then the the number is also divisible by =20( = LCM of 4 and 5 )

$\displaystyle \sf{} n(A \cap \: B) = \bigg[ \frac{700}{20} \bigg] = 35$

So The total number of numbers up to 700 are divisible by 4 or 5

$\sf{} n(A \cup \: B)$

$= \sf{}n(A) + n(B) - n(AB)$

$= \sf{}175 + 140 - 35$

$\sf{} = 280$

FINAL RESULT

The total number of numbers up to 700 are divisible by 4 or 5 is 280

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