NCERT Class X Chapter 5: Arithmetic Progression Example 7

NCERT Class X Chapter 5: Arithmetic Progression Example 7

Question:

How many two-digit numbers are divisible by 3?.

Given:

Two-digit numbers.

To Find:

The number of two-digit numbers divisible by 3.

Formula:

The number of multiples of 3 between a and b is given by \( \frac{m}{3} - \frac{n}{3} + 1 \).

Where,

m and n are numbers between a and b which are divisible by 3. 

\( m \leq b \) and
\( n \geq a \)

Solution:

The smallest two-digit number is 10. The largest two-digit number is 99.
So, a = 10 and b = 99

We need to find the number of multiples of 3 between 10 and 99.

The smallest multiple of 3 greater than or equal to 10 is 12 (3 x 4). 

The largest multiple of 3 less than or equal to 99 is 99 (3 x 33).

Therefore, m = 99 and n = 12

The number of multiples is \( \frac{99}{3} - \frac{12}{3} + 1 \) = 33 - 4 + 1 = 30.

Result:

There are 30 two-digit numbers divisible by 3.

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