Question

The number of multiples of 4 lying between 12 and 250 is :

• A. 59
• B. 59.5
• C. 60
• D. 61

Hint:

An alternate shortcut to find the number of multiples of $k$ in an open interval $(A, B)$ is:
Number of multiples $= \lfloor \frac{B - 1}{k} \rfloor - \lfloor \frac{A}{k} \rfloor$
Here, $A = 12$, $B = 250$, and $k = 4$.
Number of multiples $= \lfloor \frac{249}{4} \rfloor - \lfloor \frac{12}{4} \rfloor = 62 - 3 = 59$.
This formula is extremely fast and avoids setting up an A.P. equation entirely!

Answer 1

The Correct Option is A

Step 1: Count using division instead of building an A.P. and using the nth-term formula.
The number of multiples of $4$ that are less than or equal to any whole number $N$ is given by $\left\lfloor \frac{N}{4} \right\rfloor$, the integer part of $N$ divided by $4$. We can use this twice and subtract, instead of finding the first term, last term and applying the A.P. formula.

Step 2: Count multiples of 4 up to 249 (since we want values strictly less than 250).
\[ \left\lfloor \frac{249}{4} \right\rfloor = \left\lfloor 62.25 \right\rfloor = 62 \]
So there are $62$ multiples of $4$ from $4$ up to $248$.

Step 3: Count multiples of 4 up to 12 (since we want values strictly greater than 12, these must be excluded).
\[ \left\lfloor \frac{12}{4} \right\rfloor = 3 \]
So there are $3$ multiples of $4$ up to and including $12$, namely $4, 8, 12$.

Step 4: Subtract to remove the multiples that are not strictly between 12 and 250.
\[ 62 - 3 = 59 \]

Step 5: Sanity check the endpoints.
The first multiple of 4 strictly greater than 12 is $16$, and the last multiple of 4 strictly less than 250 is $248$. Both are accounted for correctly in our counts of 62 and 3 above.

Final Answer:
The number of multiples of 4 between 12 and 250 is $59$, which matches Option (A). \[ \boxed{59} \]