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Number Theory

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CAT 2025 Lesson : Number Theory - Counting Integers

6. Counting Consecutive Integers

This is a very simple concept which very often has direct and indirect application in questions.

If

x

and

y

are integers with

y \gt x

, the number of integers between

y

and

x

Condition	Number of Integers
where $y$ and $x$ are both to be included	$y - x + 1$
where one of $x$ and $y$ is to be included while the other is not	$y - x$
where $y$ and $x$ are both to be excluded	$y - x - 1$

Example 17

x

is an integer,

x + 23 \gt 0

and

x - 3 \lt 2

, then how many values can

x

take?

Solution

x + 23 \gt 0

⇒

x \gt -23 \longrightarrow

(a)

x - 3 \lt 2

⇒

x \lt 5 \longrightarrow

(b)

We need to count all the integers between

-23

and

5

, where

-23

and

5

are not to be included.

\therefore

Number of Integers

= 5 - (-23) - 1

= 5 + 23 - 1 = 27

Answer:

27

a_1

a_2

a_3

, … , an is an increasing AP with a common difference of

d

, then the number of terms in the AP,

- where

a_n

and

a_1

are both to be included =

\dfrac{a_n - a_1}{d} + 1

- where one is to be included while the other is not =

\dfrac{a_n - a_1}{d}

- where

a_n

and

a_1

are both to be excluded =

\dfrac{a_n - a_1}{d} - 1

Example 18

How many multiples of

7

exist between

200

and

450

Solution

When the two end points of the range,

200

and

450

, are divided by

7

, the quotient helps us identify the smallest and largest multiples of

7

in this range.

\therefore

In the range of

200

450

,

Smallest multiple of

7

7 \times 29 = 203

Largest multiple of

7

7 \times 64 = 448

The multiples in this range are

7 \times 29

7 \times 30

7 \times 31

, ... ,

7 \times 64

.

Number of multiples is the number of consecutive integers between

29

and

64

(both inclusive).

Number of multiples

= 64 - 29 + 1 = 36

Answer:

36

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