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Question #40605

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be a $3 \times 3$ matrix whose characteristic equation is $x^3 - c_1 x^2 + c_2 x - c_3 = 0$ . The roots of this equation are $\lambda_1, \lambda_2,$ and $\lambda_3$ , which are given by: $\lambda_1 = \sum_{n=1}^{60} E_2(n!), \quad \lambda_2 = \sum_{n=1}^{75} E_3(n!), \quad \lambda_3 = \sum_{n=1}^{100} E_5(n!)$ where $E_p(k)$ denotes the exponent of the highest power of prime $p$ dividing $k$ . Find the value of $c_1$ .

Options

A.
4058
B.
5032
C.
4124
D.
3986

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Exponent of Prime in n!|Characteristic Equation|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

This question tests the integration of Legendre's formula with matrix characteristic equations.

The primary trap lies in the summation of the Greatest Integer Function $\sum \lfloor n/p^k \rfloor$ .

Students often attempt to swap the summation order but fail to correctly account for the upper bounds when $N$ is not of the form $p^m - 1$ , leading to off-by-one discontinuity errors.

The optimal bypass is using the base- $p$ digit sum identity $E_p(n!) = (n - S_p(n))/(p-1)$ , which transforms a nested floor function trap into a combinatorial digit-counting problem.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Exponent of Prime in n!
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free