How Many Multiples of 4 Lie Between 10 and 250?

The question 'How many multiples of 4 lie between 10 and 250?' is one of the most frequently asked, high-scoring questions in the CBSE Class 10 Mathematics board exam. It requires using the formula of an Arithmetic Progression (AP). Let's solve it step-by-step.

Question (Click to Flip)

Can I solve this without the AP formula by just counting?

Answer

Technically yes, but it would take you 20 minutes to write down all 60 numbers during a strict 3-hour exam. The AP formula gives you the exact answer in less than 2 minutes.

Card 1 of 1 free previews

Key Facts

If an exam question asks 'between 10 and 248', remember that 'between' means you strictly cannot include the boundary numbers themselves, even if they are divisible. You would have to use 244 as the last term.

An Arithmetic Progression (AP) is any massive sequence of numbers where the difference between any two consecutive numbers is always completely identical.

Step 1: Finding the First Multiple (a)

We need numbers greater than 10 that are perfectly divisible by 4.
Let's check: 10 is not divisible by 4. 11 is not. But 12 is ($4 \times 3 = 12$).
So, the very first multiple in our AP series is 12.
First term ($a$) = 12.

Step 2: Finding the Last Multiple (l or an)

We need a number less than 250 that is perfectly divisible by 4.
If we divide 250 by 4, we get a remainder of 2.
So, we subtract the remainder from 250: $250 - 2 = 248$.
Let's check: $248 \div 4 = 62$ (Perfect division!).
So, the absolute last multiple in our series is 248.
Last term ($a_n$) = 248.

Step 3: Creating the AP and Applying the Formula

Our massive Arithmetic Progression series looks like this: 12, 16, 20, ......, 248.

First term ($a$) = 12
Common Difference ($d$) = 4
Last term ($a_n$) = 248

The famous AP formula is: $a_n = a + (n - 1)d$

Let's plug in the numbers:

$248 = 12 + (n - 1)4$
$248 - 12 = 4(n - 1)$
$236 = 4(n - 1)$
$236 \div 4 = n - 1$
$59 = n - 1$
$n = 59 + 1$
$n = 60$

Conclusion

There are exactly 60 multiples of 4 that lie between 10 and 250.

Questions and Answers

Can I solve this without the AP formula by just counting?+

Technically yes, but it would take you 20 minutes to write down all 60 numbers during a strict 3-hour exam. The AP formula gives you the exact answer in less than 2 minutes.

More in Maths

📐

Algebraic Formula for a³ + b³ + c³

Learn the important algebraic identity formula for a3 + b3 + c3. Discover the special case when a + b + c = 0 and how to solve problems.

📐

a³ + b³ + c³ Formula — Algebraic Identity

a³+b³+c³−3abc = (a+b+c)(a²+b²+c²−ab−bc−ca). Special case: if a+b+c=0, then a³+b³+c³=3abc. Full derivation and solved problems.

📐

A∩B — A Intersection B — Set Theory Explained

A∩B (A intersection B) = elements common to both sets A and B. Learn definition, Venn diagram, properties, and solved examples. Also covers A∪B (union).

📐

Formula for (a + b + c) Whole Square

Learn the standard algebraic expansion formula for (a+b+c)^2. Understand how to derive it and solve mathematical polynomial equations.

📐

Additive Inverse of 256

The additive inverse of 256 is -256. Additive inverse of any number n is -n, because n + (-n) = 0. Examples and explanation for Class 6–8 Maths.

Study Smarter with Shinyu.ai

Turn this guide into revision flashcards, a practice exam, or an AI-generated podcast — free, no signup required.