Question

Count the number of integers with in the range 100 to 400, where
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Answer 1

Answer:

Let S(n)= number of divisors of n between 100 and 400 (both included)

S(n)=(300/n) +1 if 300 is divisible by n

S(n)=floor(300/n) if 300 is not divisible by n

Using Inclusion-Exclusion Principle:

S(2|3|5|7)=S(2)+S(3)+S(5)+S(7)

-S(2&3)-S(2&5)-S(2&7)

-S(3&5)-S(3&7)

-S(5&7)

-S(2&3&5)-S(2&3&7)-S(3&5&7)

-S(2&3&5&7)

=151+101+61+42

-51-31-21

-21-14

-8

-11-7-2

-1

= 188

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Answer 2

Answer:

There are 300 integers in the range of 100 to 400.

Given:

Range is given as 100 to 400.

100, 101, 102, 103, -------------------------------------, 399

We will not consider 400 in this because it is a range of numbers and we don't consider the upper limit in a range.

Find:

The number of integers in the given range.

Solution:

The given range is,

100, 101, 102, 103, -------------------------------------, 399

We can see that this is an AP and we have to find out the value of n in this AP.

First term, a = 100

Common difference, d = a₂ - a₁

                                   d = 101 - 100 = 1

Last term, aₙ = 399                   ----------------------- (i)

but aₙ = a + (n - 1)d                    ----------------------- (ii)

Equating (i) and (ii), we get

399 = a + (n - 1)d

399 = 100 + (n - 1)(1)

399 - 100 = n - 1

299 = n - 1

n - 1 = 299

n = 299 + 1

n = 300

Hence, the number of integers in the range 100 to 400, n = 300.

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