JEE 2023 — Mathematics PYQ

Q: How do I solve JEE 2023 Mathematics questions?

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JEE Mathematics questions range from moderate to advanced difficulty. Consistent practice with official previous year papers and topic-wise drills on Mathem Solvex is the most effective preparation strategy.

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

The total number of 4-digit numbers whose greatest common divisor with 54 is 2, is:

Choose the correct answer:

Explanation

Solution

Step 1: 54 ka factorization

Humein pata hai ki:

54 = 2 \times 3^3

Step 2: Condition samajhna

$\gcd(n, 54) = 2$ hone ke liye:

$n$ ko 2 se divisible hona chahiye.
$n$ ko 3 se divisible nahi hona chahiye (kyunki agar $n$ , $3$ se cut gaya toh $\gcd$ kam se kam $2 \times 3 = 6$ ho jayega).

Iska matlab humein aise numbers chahiye jo Even hain par 6 ke multiples nahi hain.

Step 3: 2 se divisible 4-digit numbers nikaalna

Total 4-digit numbers $1000$ se $9999$ tak hote hain (Total = $9000$ ).

Inmein se 2 se divisible numbers:

n(\text{Divisible by 2}) = \frac{9000}{2} = 4500

Step 4: 6 se divisible 4-digit numbers nikaalna

(Jo 2 aur 3 dono se divisible hon):

n(\text{Divisible by 6}) = \frac{9000}{6} = 1500

Step 5: Final Calculation

Required numbers = (Total even numbers) - (Even numbers divisible by 3)

\text{Total} = 4500 - 1500

\text{Total} = 3000

Final Answer:

3000

Choose the correct answer:

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