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GRE Quantitative Arithmetic Mastery Hub: The Industry Founda

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Q1Domain Verified
According to "The Complete GRE Number Properties & Integers Course 2026: From Zero to Expert!", which of the following statements about prime factorization is MOST crucial for solving problems involving the greatest common divisor (GCD) of two large integers without explicit computation of their prime factors?
The GCD of two numbers is the product of their lowest common powers of shared prime factors.
D) If a prime number $p$ divides the product $ab$, then $p$ must divide either $a$ or $b$ (or both).
Understanding the Fundamental Theorem of Arithmetic ensures that the prime factorization of any integer greater than 1 is unique.
The GCD of two numbers can be found by repeatedly subtracting the smaller number from the larger number until a remainder of zero is achieve
Q2Domain Verified
In the context of "The Complete GRE Number Properties & Integers Course 2026: From Zero to Expert!", how does the concept of modular arithmetic, specifically congruences, provide an elegant shortcut for determining the last digit of a large power of an integer, even when direct calculation is infeasible?
By using Fermat's Little Theorem to reduce the exponent modulo a prime number that divides 10.
By focusing on the cyclical nature of remainders when powers of the base are divided by 10, leveraging the property $a \equiv b \pmod{m}$ implies $a^n \equiv b^n \pmod{m}$.
By finding the prime factorization of the base and applying the Chinese Remainder Theorem to the congruences modulo each prime factor of 10.
By expressing the base as a sum of its digits and applying the congruence property $a \equiv b \pmod{10}$ implies $a^n \equiv b^n \pmod{10}$.
Q3Domain Verified
"The Complete GRE Number Properties & Integers Course 2026: From Zero to Expert!" emphasizes the importance of understanding divisibility rules. When faced with a large integer $N$ and asked to determine if it's divisible by 7, which of the following applications of number properties offers the most robust and universally applicable method, superior to simple trial division?
Applying the divisibility rule for 7 which involves doubling the last digit and subtracting it from the remaining number, repeating the process until a small number is reached.
Decomposing $N$ into its prime factors and checking if 7 is one of those factors.
Repeatedly subtracting multiples of 7 from $N$ until a remainder of 0 or a number between 1 and 6 is obtained.
Expressing $N$ in terms of powers of 10 and using modular arithmetic properties, specifically congruences modulo 7, to simplify the divisibility check.

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