DSSSB PGT Arithmetical & Numerical Ability Mastery Hub: Prac

In the context of the given course, what is the primary characteristic that distinguishes a "perfect number" from a "deficient number"?

probes the fundamental definition of perfect and deficient numbers, a core concept in number theory discussed in the course. Option A accurately defines both terms: a perfect number's proper divisors sum to the number itself (e.g., 6 = 1+2+3), and a deficient number's proper divisors sum to less than the number (e.g., 8: 1+2+4 = 7 < 8). Option B is incorrect because while the first few perfect numbers are even, it's an open conjecture whether odd perfect numbers exist. Option C introduces an irrelevant criterion; divisibility by primes up to the square root is related to primality testing, not the definition of perfect or deficient numbers. Option D is incorrect; perfect numbers can have varying numbers of prime factors (e.g., 6 = 2*3 has two, but 28 = 2^2 * 7 has two prime factors but three distinct divisors including the number itself. The definition hinges on the sum of divisors, not the count of prime factors. Question: Consider the set of integers $Z$. If $a \equiv b \pmod{m}$ and $a \equiv c \pmod{m}$, which of the following statements is a direct consequence of the properties of modular arithmetic as taught in the course?

tests the understanding of the transitive property of modular congruence and its implications. If $a \equiv b \pmod{m}$, it means $a-b = km$ for some integer $k$. If $a \equiv c \pmod{m}$, it means $a-c = lm$ for some integer $l$. Subtracting these equations, $(a-b) - (a-c) = km - lm$, which simplifies to $c-b = (k-l)m$. This directly implies $b-c = -(k-l)m$, meaning $b-c$ is a multiple of $m$. Therefore, $b \equiv c \pmod{m}$. Option A is partially correct in stating $b \equiv c \pmod{m}$, but the second part, $m \equiv 0 \pmod{b-c}$, is a restatement of the same relationship and not a distinct consequence. Option B correctly states $b \equiv c \pmod{m}$ and $b-c$ is a multiple of $m$. Option C correctly states $b \equiv c \pmod{m}$ and $m$ is a divisor of $b-c$, which is the definition of $b \equiv c \pmod{m}$. Since Options B and C are both valid and D encompasses all correct implications derived from the transitive property and definition of modular arithmetic, D is the most comprehensive and correct answer. Question: In the advanced divisibility rules discussed in the course, what is the underlying principle that allows for determining the divisibility of a number by 7 using alternating sums of blocks of three digits?