JEE Main 2023 (29 January Shift 2)

LEVELJEE Main

Animated Solution for Mathematics - Permutations and Combinations: The total number of 4-digit numbers whose greatest common divisor with 54 is 2, is ____.

Q: Wait, why do we reject multiples of 3? Can't the GCD just be 2 even if it's a multiple of 3?

That is a fantastic question! If $N$ is a multiple of 3, then $N$ shares a factor of 3 with 54. Since 54 is $2 \times 3^3$, if $N$ is also even, the GCD would be at least $\gcd(6, 54) = 6$. Since we require the GCD to be exactly 2, we cannot allow any factor of 3 to exist in $N$.

Enter Numerical Value:

Visualized Solution

The Problem Statement

Find the total number of 4-digit numbers $N$ .
Condition: $g cd (N, 54) = 2$ .

Prime Factorization of $54$

Let's break down $54$ into its prime factors.
$54 = 2 \times 27 = 2^{1} \times 3^{3}$ .

Decoding the GCD Condition

Given: $g cd (N, 2 \times 3^{3}) = 2$ .
This means $N$ must have a factor of $2$ (i.e., $N$ is even).
$N$ must not have a factor of $3$ (otherwise GCD would be $\geq 6$ ).

The Universe of 4-Digit Numbers

Smallest 4-digit number $= 1000$ .
Largest 4-digit number $= 9999$ .
Total numbers $= 9999 - 1000 + 1 = 9000$ .

Numbers Divisible by $2$

Let Set $A$ be the numbers divisible by $2$ .
Every alternate number is even.
$n (A) = \frac{9000}{2} = 4500$ .

The Forbidden Multiples

Let Set $B$ be the numbers divisible by $3$ .
We must exclude numbers that are in both Set $A$ and Set $B$ .
These are numbers divisible by both $2$ and $3$ .

Numbers Divisible by Both $2$ and $3$

Numbers divisible by both $2$ and $3$ are multiples of $LCM (2, 3) = 6$ .
$n (A \cap B) = \frac{9000}{6} = 1500$ .

Calculating the Required Region

Required numbers = (Divisible by $2$ ) - (Divisible by $6$ ).
Required $= n (A) - n (A \cap B)$ .
Required $= 4500 - 1500 = 3000$ .

Final Answer

The total number of such 4-digit numbers is $3000$ .
Key Takeaway: $g cd (N, A) = k ⟹ k$ divides $N$ , and $g cd (\frac{N}{k}, \frac{A}{k}) = 1$ .

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The Sigma Insight: Fundamental Principle of Counting

Solution Diagram

Analyzing the Prime Structure

To solve for the total number of 4-digit integers

N

such that

g cd (N, 54) = 2

, we first perform the prime factorization of 54:

54 = 2 \times 27 = 2^{1} \times 3^{3}

For the condition

g cd (N, 54) = 2

to hold,

N

must satisfy two specific constraints: 1.

N

must be a multiple of 2. 2.

N

must not be a multiple of 3.

N

were a multiple of 3, the

g cd (N, 54)

would be at least 6, which contradicts our requirement.

Defining the Universe

We are considering 4-digit integers, which range from 1000 to 9999. The total number of integers in this set is:

Total = 9999 - 1000 + 1 = 9000

Within this range, we first isolate the set of even numbers. Since every second number is even, the count of even numbers is:

n (Even) = \frac{9000}{2} = 4500

The Set Theory Battle

We must now remove the "troublemakers"—the numbers that are multiples of 3. Since we are restricted to the set of even numbers, the numbers that are both even and multiples of 3 are precisely the multiples of 6.

We calculate the number of multiples of 6 within our range of 9000 integers:

n (Multiples of 6) = \frac{9000}{6} = 1500

Final Calculation

To find the count of numbers that are even but not divisible by 3, we subtract the count of multiples of 6 from the total count of even numbers:

Result = 4500 - 1500 = 3000

The total number of 4-digit integers

N

such that

g cd (N, 54) = 2

is 3000.

Common Traps & FAQs

Wait, why do we reject multiples of 3? Can't the GCD just be 2 even if it's a multiple of 3?

That is a fantastic question! If

N

is a multiple of 3, then

N

shares a factor of 3 with 54. Since 54 is

2 \times 3^{3}

, if

N

is also even, the GCD would be at least

g cd (6, 54) = 6

. Since we require the GCD to be exactly 2, we cannot allow any factor of 3 to exist in

N

Why do we subtract multiples of 6? I thought we were just looking for even numbers.

We are looking for even numbers that are NOT multiples of 3. The set of 'even numbers' includes numbers like 6, 12, 18, etc., which are multiples of 3. These are the 'troublemakers' that violate our condition. By subtracting the multiples of 6 (which are the intersection of 'even' and 'multiple of 3'), we effectively filter out all the numbers that contain a factor of 3.

Is there a faster way to do this using modular arithmetic?

You could use the Chinese Remainder Theorem or modular constraints, but for a range of 9000 numbers, the set theory approach (Inclusion-Exclusion Principle) is the most robust and least prone to calculation errors. Stick to the set logic; it is the most reliable path to the correct answer.

Animated Solution for Mathematics - Permutations and Combinations: The total number of 4-digit numbers whose greatest common divisor with 54 is 2, is ____.

Enter Numerical Value:

Visualized Solution

The Problem Statement

Prime Factorization of $54$

Decoding the GCD Condition

The Universe of 4-Digit Numbers

Numbers Divisible by $2$

The Forbidden Multiples

Numbers Divisible by Both $2$ and $3$

Calculating the Required Region

Final Answer

The Sigma Insight: Fundamental Principle of Counting

Analyzing the Prime Structure

Defining the Universe

The Set Theory Battle

Final Calculation

Similar Questions

The total number of four digit numbers such that each of the first three digits is divisible by the last digit, is equal to ______.

The number of natural numbers lying between 1012 and 23421 that can be formed using the digits 2, 3, 4, 5, 6 (repetition of digits is not allowed) and divisible by 55 is ______.

Total numbers of 3-digit numbers that are divisible by 6 and can be formed by using the digits 1, 2, 3, 4, 5 with repetition, is ________

The number of 5 digit numbers which are divisible by 4, with digits from the set ${1, 2, 3, 4, 5}$ and the repetition of digits is allowed, is ________.

Numbers are to be formed between 1000 and 3000, which are divisible by 4, using the digits 1, 2, 3, 4, 5 and 6 without repetition of digits. Then the total number of such numbers is ______.

Let $m$ and $n, (m < n)$ be two 2-digit numbers. Then the total numbers of pairs $(m, n)$ , such that $g cd (m, n) = 6$ , is ________

Number of 4-digit numbers (the repetition of digits is allowed) which are made using the digits 1, 2, 3 and 5, and are divisible by 15, is equal to ____.

The number of 4-digit integers in the closed interval $[2022, 4482]$ formed by using the digits 0, 2, 3, 4, 6, 7 is _____________.

The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ......... .

The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1, 2, 3, 4, 5, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5 is

Animated Solution for Mathematics - Permutations and Combinations: The total number of 4-digit numbers whose greatest common divisor with 54 is 2, is ____.

Enter Numerical Value:

Visualized Solution

The Problem Statement

Prime Factorization of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 54

Decoding the GCD Condition

The Universe of 4-Digit Numbers

Numbers Divisible by .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 2

The Forbidden Multiples

Numbers Divisible by Both .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 2 and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 3

Calculating the Required Region

Final Answer

The Sigma Insight: Fundamental Principle of Counting

Analyzing the Prime Structure

Defining the Universe

The Set Theory Battle

Final Calculation

Similar Questions

The total number of four digit numbers such that each of the first three digits is divisible by the last digit, is equal to ______.

The number of natural numbers lying between 1012 and 23421 that can be formed using the digits 2, 3, 4, 5, 6 (repetition of digits is not allowed) and divisible by 55 is ______.

Total numbers of 3-digit numbers that are divisible by 6 and can be formed by using the digits 1, 2, 3, 4, 5 with repetition, is ________

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of 5 digit numbers which are divisible by 4, with digits from the set {1,2,3,4,5} and the repetition of digits is allowed, is ________.

Numbers are to be formed between 1000 and 3000, which are divisible by 4, using the digits 1, 2, 3, 4, 5 and 6 without repetition of digits. Then the total number of such numbers is ______.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let m and n,(m<n) be two 2-digit numbers. Then the total numbers of pairs (m,n), such that gcd(m,n)=6, is ________

Number of 4-digit numbers (the repetition of digits is allowed) which are made using the digits 1, 2, 3 and 5, and are divisible by 15, is equal to ____.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of 4-digit integers in the closed interval [2022,4482] formed by using the digits 0, 2, 3, 4, 6, 7 is _____________.

The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ......... .

The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1, 2, 3, 4, 5, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5 is

Prime Factorization of $54$

Numbers Divisible by $2$

Numbers Divisible by Both $2$ and $3$

The number of 5 digit numbers which are divisible by 4, with digits from the set ${1, 2, 3, 4, 5}$ and the repetition of digits is allowed, is ________.

Let $m$ and $n, (m < n)$ be two 2-digit numbers. Then the total numbers of pairs $(m, n)$ , such that $g cd (m, n) = 6$ , is ________

The number of 4-digit integers in the closed interval $[2022, 4482]$ formed by using the digits 0, 2, 3, 4, 6, 7 is _____________.