How many multiples of 4 lie between 10 and 250? (Hint: All multiples of 4 form an AP. Find the smallest and largest multiples of 4 between 10 and 250.)

Question

Homi · Accepted Answer

Understand the Question

The multiples of $4$ strictly between $10$ and $250$ form an Arithmetic Progression (AP) with a common difference of $d = 4$ .
The first multiple of $4$ greater than $10$ is $12$ , and the largest multiple of $4$ less than $250$ is $248$ .
Using the $n^{\text{th}}$ term formula $a_n = a + (n - 1)d$ , we can solve for the total number of terms $n$ .

Step 1 · Identify the AP and its terms

The multiples of $4$ lying between $10$ and $250$ are: $12, 16, 20, \dots, 248$

This forms an Arithmetic Progression (AP) where:

First term, $a = 12$
Common difference, $d = 16 - 12 = 4$
Last term, $a_n = 248$

Step 2 · Find the number of terms

The $n^{\text{th}}$ term of an AP is given by: $a_n = a + (n - 1)d$

Substitute the values:

\begin{aligned} 248 &= 12 + (n - 1)4 \\[0.6em] 248 - 12 &= 4(n - 1) \\[0.6em] 236 &= 4(n - 1) \\[0.6em] n - 1 &= \dfrac{236}{4} \\[0.6em] n - 1 &= 59 \\[0.6em] n &= 59 + 1 = 60 \end{aligned}

Answer

60

Common Mistakes

Including Non-multiples as Boundaries: Taking $10$ or $250$ directly as terms of the AP without verifying divisibility by $4$ .
Off-by-One Error: Forgetting to add $1$ after dividing $\dfrac{236}{4} = 59$ , leading to an incorrect answer of $59$ instead of $60$ .

Question 4

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