Question

There are certain number of positive integers, which are
multiple of 15 and lie between 100 and 1 Lakh Of these how
many are multiple of 35​

Answer 1

Answer:

there are ten

Step-by-step explanation:

35,70,105,140,175,210,245,280,315,350

Answer 2

Answer:

The number of  multiples of 15 between 100 and 1 lakh, which are also multiple of 35 = 952

Step-by-step explanation:

To find,

The number positive integers which are multiples both 15 and 35 between 100 and 1 lakh

Recall the formula

The number of terms in an AP

n = $\frac{t_n - t_1}{d} +1$, -----------------(1)

where tₙ is the last term and t₁ is the first term and d is the common difference of the AP

Solution:

To find LCM(15,35)

Prime factorization of 15 = 3×5

Prime factorization of 35 = 5×7

LCM of 15 and 35 = 3×5×7 = 105

Hence multiples of 15 which are also multiples of 35 are the multiples of 105

Now, we need to calculate the number of multiples of 105 between 100 and 1 lakh

The first multiple of 105 between 100 and 1lakh = 105

To find the last multiple of 105 between 100 and 1lakh

On dividing 1 lakh by 105, we get

quotient = 952 and remainder = 40

Then we can write,

100000 = 952×105 + 40

100000 - 40 = 952×105, a multiple of 105

Hence the last multiple of 105, between 100 and 1 lakh = 99960

The multiples of 105 between 100 and 1lakh are

105,210,315,..............., 99960

This sequence forms an AP and is required to find the number of terms in this AP

Here first term  t₁ = 105

last term tₙ = 99960

common difference =d = 105

Then from equation (1), the  number of terms of the AP  =  $\frac{99960 - 105}{105} +1$

= $\frac{99855}{105} +1$

= 951 +1

= 952

∴ The number of  multiples of 15 between 100 and 1 lakh, which are also multiple of 35 = 952

#SPJ2