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CAT 2025 Lesson : Number Theory - Counting Integers

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6. Counting Consecutive Integers

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

If
xx and yy are integers with y>xy \gt x, the number of integers between yy and xx,

Condition Number of Integers
where yy and xx are both to be included yx+1y - x + 1
where one of xx and yy is to be included while the other is not yxy - x
where yy and xx are both to be excluded yx1y - x - 1


Example 17

If xx is an integer, x+23>0x + 23 \gt 0 and x3<2x - 3 \lt 2, then how many values can xx take?

Solution

x+23>0x + 23 \gt 0
x>23x \gt -23 \longrightarrow (a)

x3<2x - 3 \lt 2
x<5x \lt 5 \longrightarrow (b)

We need to count all the integers between
23-23 and 55, where 23-23 and 55 are not to be included.

\therefore Number of Integers =5(23)1= 5 - (-23) - 1
=5+231=27= 5 + 23 - 1 = 27

Answer:
2727


If
a1a_1, a2a_2, a3a_3, … , an is an increasing AP with a common difference of dd, then the number of terms in the AP,

- where
ana_n and a1a_1 are both to be included                    = ana1d+1\dfrac{a_n - a_1}{d} + 1
- where one is to be included while the other is not    = 
ana1d\dfrac{a_n - a_1}{d}
- where
ana_n and a1a_1 are both to be excluded                   = ana1d1\dfrac{a_n - a_1}{d} - 1

Example 18

How many multiples of 77 exist between 200200 and 450450?

Solution

When the two end points of the range, 200200 and 450450, are divided by 77, the quotient helps us identify the smallest and largest multiples of 77 in this range.

\therefore In the range of 200200 to 450450,

Smallest multiple of
77 = 7×29=2037 \times 29 = 203
Largest multiple of
77 = 7×64=4487 \times 64 = 448

The multiples in this range are
7×297 \times 29, 7×307 \times 30, 7×317 \times 31, ... , 7×647 \times 64.

Number of multiples is the number of consecutive integers between
2929 and 6464 (both inclusive).

Number of multiples
=6429+1=36= 64 - 29 + 1 = 36

Answer:
3636

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