Question

How many integers between 100 and 1500(both inclusive) are divisible by 5 or 11??

Answer 1

Answer:

Answer is 1400 and yes it is divisible by 05

Answer 2

Answer:

The number of integers between 100 and 1500 (both inclusive), that are divisible by 5 or 11 = 382

Step-by-step explanation:

To find,

The number of integers between 100 and 1500 (both inclusive), that are divisible by 5 or 11

The number of integers divisible by  5 or 11 = The number of integers divisible by 5 + The number of integers divisible by 11 - The number of integers divisible 5 and 11 ----------------(1)

Recall the formula

The number of terms of an AP is given by the formula

n = $\frac{Last \ term - first \ term}{common \ difference } +1$

The number of integers divisible by 5

The integers divisible by 5  between 100 and 1500 (both inclusive) are

100,105,110,-------1500

This forms an AP with first term = 100

Last term = 1500

common difference = 5

Hence, the number of terms in this AP = $\frac{1500 - 100}{5} +1$

= 280 + 1

= 281

There are 281 numbers between 100 and 1500 including 100 and 1500 that are divisible by 5

The number of integers divisible by 11

The integers divisible by 11  between 100 and 1500 (both inclusive) are

110,121,132,-------1496

This forms an AP with first term = 110

Last term = 1496

common difference = 11

Hence, the number of terms in this AP = $\frac{1496 - 110}{11} +1$

= 126 + 1

= 127

There are 127 numbers between 100 and 1500 including 100 and 1500 that are divisible by 11

The number of integers divisible by 5 and 11

The number of integers divisible by 5 and 11 = The number of integers divisible by the LCM(5,11)

Since 5 and 11 are coprime, LCM (5,11) = 55

The integers divisible by 55  between 100 and 1500 (both inclusive) are

110, 165,220,............., 1485

This forms an AP with first term = 110

Last term = 1485

common difference = 55

Hence, the number of terms in this AP = $\frac{1485 - 110}{55} +1$

=  25+ 1

= 26

There are 26 numbers between 100 and 1500 including 100 and 1500 that are divisible by 55

Then, from equation(1) we have

The number of integers divisible by  5 or 11 = 281+127 - 26

= 382

∴The number of integers between 100 and 1500 (both inclusive), that are divisible by 5 or 11 = 382

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