Question

How many numbers between 200 and 500 are completely divisible by 3 ,4 and 5?​

Answer 1

Given:

Range of numbers = 200 to 500

To find:

How many numbers between 200 and 500 are completely divisible by 3,4 and 5.

Solution:

Since the numbers between 200 and 500 should be divided by all the three numbers i.e. 3, 4 and 5.

We would have to take LCM of 3, 4 and 5 which is 60.

The first number that would be divisible by 60 after 200 is 240.

Adding 60 to each number we get the series: 240, 300, 360, 420, 480, 540 and so on.

Since we only want numbers up till 500, we would only have 240, 300, 360, 420, 480, which are 5 in number.

Therefore 5 numbers are divisible by 3, 4 and 5 between 200 and 500.

Answer 2

SOLUTION

TO DETERMINE

The number of numbers between 200 and 500 are completely divisible by 3 ,4 and 5

EVALUATION

A number which is divisible by 3 ,4 and 5 , then the number is also divisible by their LCM

LCM of 3 ,4 and 5 = 3 × 4 × 5 = 60

Now the numbers between 200 and 500 are completely divisible by 3 ,4 and 5 form an arithmetic progression 240, 300,......., 480

First term = a = 240

Common Difference = d = 300 - 240 = 60

Let 480 be the nth term of the Arithmetic progression

$\therefore \: \sf{a + (n - 1)d = 480}$

$\\ \implies \: \sf{240 +60 (n - 1) = 480}$

$\\ \implies \: \sf{60 (n - 1) = 240}$

$\\ \implies \: \sf{ (n - 1) = 4}$

$\\ \implies \: \sf{n = 5}$

FINAL ANSWER

The number of numbers between 200 and 500 are completely divisible by 3 ,4 and 5 is 5

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