Class JEE Mathematics Sets, Relations, and Functions Q #1081
KNOWLEDGE BASED
APPLY
4 Marks 2023 JEE Main 2023 (Online) 15th April Morning Shift NUMERICAL
The number of elements in the set $\left\{n \in \mathbb{N}: 10 \leq n \leq 100\right.$ and $3^{n}-3$ is a multiple of 7$\}$ is ___________.
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Step-by-Step Solution

  1. First, we need to find the values of $n$ for which $3^n - 3$ is a multiple of 7. This means $3^n - 3 \equiv 0 \pmod{7}$, or $3^n \equiv 3 \pmod{7}$.
  2. We can simplify this to $3^{n-1} \equiv 1 \pmod{7}$.
  3. Now, let's find the powers of 3 modulo 7:
    • $3^1 \equiv 3 \pmod{7}$
    • $3^2 \equiv 2 \pmod{7}$
    • $3^3 \equiv 6 \pmod{7}$
    • $3^4 \equiv 4 \pmod{7}$
    • $3^5 \equiv 5 \pmod{7}$
    • $3^6 \equiv 1 \pmod{7}$
    So, the order of 3 modulo 7 is 6. This means $3^k \equiv 1 \pmod{7}$ if and only if $k$ is a multiple of 6.
  4. Therefore, $n-1$ must be a multiple of 6, i.e., $n-1 = 6k$ for some integer $k$. This gives $n = 6k + 1$.
  5. We are given that $10 \leq n \leq 100$. Substituting $n = 6k + 1$, we get $10 \leq 6k + 1 \leq 100$.
  6. Subtracting 1 from all parts of the inequality, we have $9 \leq 6k \leq 99$.
  7. Dividing by 6, we get $\frac{9}{6} \leq k \leq \frac{99}{6}$, which simplifies to $1.5 \leq k \leq 16.5$.
  8. Since $k$ must be an integer, the possible values of $k$ are $2, 3, 4, \dots, 16$.
  9. The number of integers in this range is $16 - 2 + 1 = 15$.

Correct Answer: 15

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the concepts of modular arithmetic and number theory to determine when $3^n - 3$ is divisible by 7. This requires using the properties of exponents and remainders to solve the problem.
Knowledge Dimension: PROCEDURAL
Justification: The question requires a specific procedure to find the values of 'n' that satisfy the given condition. This involves calculating powers of 3 modulo 7 and identifying a pattern.
Syllabus Audit: In the context of JEE, this is classified as KNOWLEDGE. The question tests the student's understanding of number theory concepts, specifically modular arithmetic, which is a standard topic in the JEE syllabus.

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