Question

how Many natural number in between 100and 500 multiply by 3 and 5

Answer 1

• A number which is divisible by 3,5 is their Multiple.

• Hence, it must be divisible by their Least Common Multiple(LCM) i.e 15.

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We have to find numbers which are :-

1. Between 100 and 500
2. Divisible by 15

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To do this, we need an Arithmetic Progression(AP)

First Term (a) = ?

$\textbf{Last \: Term \: (a_l) = ?}$

Common Difference (d) = 15

Number of Terms(n) = ?

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We can find First Term and Last Term by Simple Division.

Last Term :

• Divide 500 by 15.

• If there's no remainder then 500 is divisible by 15 and If there's a remainder then the number obtained by subtracting remainder from 500 is divisible by 15.

• In this case, When we divide 500 by 15, we get remainder as 5.

• Subtracting 5 from 500, we get.......... 495.
• So, Last Term is 495.

First Term :

This method is done in 2 ways. I would be doing it through the method which is simple but 1 step longer.

• Divide 100 by 15.
• If it is divisible then it is First Term and If it ain't then we have 2 methods to calculate that number.

• When we Divide 100 by 15, we get remainder as 10.
• The simple method says just subtract 10 from 100 and then add 15 to it.

       =>(100-10+15 = 105)

• Tricky method is confusing unless explained person to person so I am not explaining it to you here.
• First Term(a) is 105.

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Now,

a = 105

$a_l = 495$

d = 15

n = ?

The multiples between 100 and 500 = Number of Terms = n

Applying the Formula,

$\textbf{\boxed{a_l = a + (n-1)d}}$

=> 495 = 105 + (n-1)15

=> 15(n-1) = 495-195

=> 15(n-1) = 300

=> $n-1 = \frac{300}{15}$

=>  n-1 = 20

=> n = 21

Hence, 21 Natural numbers between 100 and 500 are divisible by 3 and 5.

OR

are their multiples.

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If you have any doubts, then feel free to put them in the comments....