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Advanced Quantitative Apt Practice Test 2026 | Exam Prep
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Q1Domain Verified
In the context of advanced number systems, consider the representation of a number in base $b$. If a number $N$ is represented as $(a_n a_{n-1} \dots a_1 a_0)_b$, and we are given that $N \equiv 0 \pmod{b-1}$, which of the following properties must hold for its digits $a_i$?
The product of the digits, $\prod_{i=0}^n a_i$, must be divisible by $b-1$.
The number formed by the last two digits, $(a_1 a_0)_b$, must be divisible by $b-1$.
The sum of the digits, $\sum_{i=0}^n a_i$, must be divisible by $b-1$.
The alternating sum of the digits, $\sum_{i=0}^n (-1)^i a_i$, must be divisible by $b-1$.
Q2Domain Verified
Consider a number system with an arbitrary base $b > 2$. If we define a "generalized Fibonacci sequence" where $F_0 = 0$, $F_1 = 1$, and $F_k = x F_{k-1} + y F_{k-2}$ for $k \ge 2$, where $x$ and $y$ are integers and $x^2 - 4y \neq 0$. What is the closed-form expression for the $n$-th term $F_n$ in terms of $b$, assuming the characteristic equation has roots $\alpha$ and $\beta$?
$F_n = \frac{\alpha^n - \beta^n}{\alpha - \beta} \pmod{b}$
$F_n = \frac{\alpha^{n+1} - \beta^{n+1}}{\alpha - \beta}$
$F_n = \frac{\alpha^n - \beta^n}{\alpha - \beta}$
$F_n = A \alpha^n + B \beta^n$, where $A$ and $B$ are constants determined by initial conditions.
Q3Domain Verified
is about a generalized Fibonacci sequence with arbitrary $x$ and $y$. Option D incorrectly introduces a modulo operation. The question asks for the closed-form expression for $F_n$, not its representation in a specific base or modulo a number. The closed-form expression is an algebraic formul
Question: In the realm of advanced number theory and its application in computational algorithms, consider the concept of the p-adic valuation, denoted by $v_p(n)$. For a prime $p$, $v_p(n)$ is the exponent of the highest power of $p$ that divides $n$. If we are given Legendre's formula for $v_p(n!)$: $v_p(n!) = \sum_{k=1}^{\infty} \lfloor \frac{n}{p^k} \rfloor$. Now, consider the binomial coefficient $\binom{2n}{n}$. Which of the following statements about $v_p\left(\binom{2n}{n}\right)$ is generally true for any prime $p$? A) $v_p\left(\binom{2n}{n}\right) = \sum_{k=1}^{\infty} \left( \lfloor \frac{2n}{p^k} \rfloor - 2 \lfloor \frac{n}{p^k} \rfloor \right)$
$v_p\left(\binom{2n}{n}\right) = \sum_{k=1}^{\infty} \left( 2 \lfloor \frac{n}{p^k} \rfloor - \lfloor \frac{2n}{p^k} \rfloor \right)$
$v_p\left(\binom{2n}{n}\right) = \sum_{k=1}^{\infty} \left( \lfloor \frac{n}{p^k} \rfloor - \lfloor \frac{2n}{p^k} \rfloor \right)$
$v_p\left(\binom{2n}{n}\right) = \sum_{k=1}^{\infty} \left( \lfloor \frac{2n}{p^k} \rfloor - \lfloor \frac{n}{p^k} \rfloor \right)$
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This domain protocol is rigorously covered in our 2026 Elite Framework. Every mock reflects direct alignment with the official assessment criteria to eliminate performance gaps.
This domain protocol is rigorously covered in our 2026 Elite Framework. Every mock reflects direct alignment with the official assessment criteria to eliminate performance gaps.
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