Question

Number of natural numbers between 100 and 500 which are multiple of 3 and 5

Answer 1

The smallest number which is divisible by 3 and 5 between 100 and 500 is 105.

The largest number which is divisible by 3 and 5 between 100 and 500 is 495.

From the properties of arithmetic progressions we know that aₓ = a + ( x - 1 ) d [ where x is the number of terms and aₓ is the value of x th term ]

Let 495 be the nth term and 105 be the first term of this given AP

⇒ 495 = 105 + ( n - 1 )d

As number should be the multiples of 3 and 5, common difference or d is  3 x 5 = 15

⇒ 495 = 105 ( n - 1 )15

⇒ 495 - 105 = ( n - 1 )15

⇒ 390 = ( n - 1 )15

⇒ $\dfrac{390}{15} = n - 1$

⇒ 26 = n - 1

⇒ 26 + 1 = n

⇒ 27 = n

As 495 is the last term of the AP, place of 495 will be the last term of the AP.

Therefore, number of natural numbers between 100 and 500 which are the multiple of 3 and 5 is 27.