What are the rules for zero and negative exponents?

• For any non-zero x: x^0 = 1 (0^0 is undefined). • x^(-n) = 1 / (x^n). • (a/b)^(-n) = (b/a)^n.

What are the common expansions for (a + b)^2 and (a - b)^2, and what common trap must be avoided?

(a + b)^2 = a^2 + 2ab + b^2 (a - b)^2 = a^2 - 2ab + b^2 Trap: (a + b)^2 is NOT equal to a^2 + b^2! The middle cross-term ±2ab is missing.

What is the GMAT divisibility rule for 6?

An integer is divisible by 6 if and only if it satisfies both divisibility by 2 (the units digit is even: 0, 2, 4, 6, 8) and divisibility by 3 (the sum of its digits is a multiple of 3).

What are Vieta's formulas for the sum and product of quadratic roots?

For ax^2 + bx + c = 0 with roots r1 and r2: • Sum of roots: r1 + r2 = -b / a • Product of roots: r1 * r2 = c / a.

What is the GMAT divisibility rule for 3 and 9?

An integer is divisible by 3 if the sum of its digits is divisible by 3. An integer is divisible by 9 if the sum of its digits is divisible by 9.

When is the reciprocal inequality rule (if x > y, then 1/x < 1/y) valid?

Only when x and y have the SAME sign (both positive or both negative). If x and y have opposite signs (e.g., x = 2 and y = -3), x > y but 1/x = 1/2 > -1/3 = 1/y (the inequality does NOT flip).

What algebraic identity defines the difference of squares and why is it tested so frequently?

a^2 - b^2 = (a - b)(a + b). It appears in disguise, such as x^4 - 16 = (x^2 - 4)(x^2 + 4) = (x - 2)(x + 2)(x^2 + 4), and is frequently used to simplify fractions without long division.

What is the parity rule for the product of integers?

Even * Anything = Even; Odd * Odd = Odd. The product of any number of integers is Odd if and only if EVERY factor in the product is Odd.

When does a fully reduced fraction a/b produce a terminating decimal?

A fraction in lowest terms terminates if and only if the prime factorization of its denominator b contains NO prime factors other than 2 and/or 5 (e.g., b = 2^m * 5^n). Any other prime factor (3, 7, 11, etc.) results in a repeating decimal.

What is the relationship between the GCD, LCM, and two positive integers a and b?

GCD(a, b) * LCM(a, b) = a * b. Note: This product rule holds strictly for two numbers, not necessarily for three or more.

What does the quadratic discriminant b^2 - 4ac determine about roots?

For ax^2 + bx + c = 0: • b^2 - 4ac > 0: Two distinct real roots • b^2 - 4ac = 0: Exactly one real root (a double root) • b^2 - 4ac < 0: No real roots (the parabola does not cross the x-axis).

When is the sum of n consecutive integers divisible by n?

• If n is ODD, the sum is always a multiple of n (Sum = n * Middle term). • If n is EVEN, the sum is NEVER a multiple of n (Sum = n * (Average of two middle terms), yielding an odd multiple of n/2).

How do you factor the sum and difference of cubes?

• Difference of cubes: a^3 - b^3 = (a - b)(a^2 + ab + b^2) • Sum of cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2) Mnemonic 'SOAP': Same sign, Opposite sign, Always Positive for the last term.

How do you calculate the total number of positive factors (divisors) for an integer N > 1?

Write the prime factorization of N using distinct primes: N = (p1^a)(p2^b)(p3^c)... Then multiply the incremented exponents: (a + 1)(b + 1)(c + 1)... For example, 72 = 2^3 * 3^2 has (3 + 1)(2 + 1) = 12 positive divisors.

When does a linear system a1*x + b1*y = c1 and a2*x + b2*y = c2 have infinitely many, zero, or one solution?

Assuming non-zero coefficients a2, b2, c2: • Unique solution (intersecting lines): a1/a2 != b1/b2. • No solution (distinct parallel lines): a1/a2 = b1/b2 != c1/c2. • Infinitely many solutions (coincident lines): a1/a2 = b1/b2 = c1/c2.

What are the parity rules for the sum and difference of integers?

Even ± Even = Even; Odd ± Odd = Even; Even ± Odd = Odd. The sum or difference is Odd if and only if the two integers have different parities.

What is the formal definition of the absolute value function |x| and the principal square root sqrt(x^2)?

|x| = x when x >= 0, and |x| = -x when x < 0. Crucial GMAT rule: sqrt(x^2) = |x|, NOT simply x. If x = -5, sqrt((-5)^2) = sqrt(25) = 5 = -(-5).

On the GMAT, does the radical sign sqrt(x) represent positive, negative, or both values?

The radical symbol sqrt(k) ALWAYS represents the non-negative principal square root. sqrt(16) = 4, NOT ±4. Only if an equation says x^2 = 16 do you have two solutions: x = +4 or x = -4.

What is the formula for the sum of the first n positive odd integers?

Sum = n^2. Example: 1 + 3 + 5 + 7 = 4^2 = 16. The average of the first n odd positive integers is n.

What are the common GMAT traps regarding prime numbers?

1. 2 is the smallest prime number and the ONLY even prime. 2. 1 is NEITHER prime nor composite. 3. Negative integers cannot be prime (primes are positive integers > 1). 4. All primes greater than 3 can be expressed as 6k ± 1 (though not all 6k ± 1 are prime).

How do you represent a positive integer N that leaves the same remainder r when divided by both a and b?

If N leaves remainder r when divided by positive integers a and b (with 0 = 0. Example: Remainder 2 when divided by 4 and 6 gives N = 12k + 2.

What is the GMAT divisibility rule for 4 and 8?

An integer is divisible by 4 if its last two digits form a number divisible by 4 (e.g., 5,324). An integer is divisible by 8 if its last three digits form a number divisible by 8 (e.g., 17,816).

For what values of x is x^2 > x?

x^2 > x is true when x > 1 OR x < 0. It is FALSE when 0 <= x <= 1 (e.g., (1/2)^2 = 1/4 < 1/2; 0^2 = 0; 1^2 = 1). Testing the (0, 1) boundary is high-yield.

How are fractional exponents x^(a/b) defined in real-number algebra?

For a positive real base x > 0 and integers a, b with b > 0, x^(a/b) = (b-th root of x)^a = b-th root of (x^a) (and for negative real x when b is odd and a/b is reduced). Example: 8^(2/3) = (cube root of 8)^2 = 2^2 = 4.

What is the units-digit cycle pattern for positive integer powers of 2, 3, 7, and 8?

All four have a repeating units-digit period of 4 for positive integer exponents n >= 1: • 2^n: 2, 4, 8, 6 • 3^n: 3, 9, 7, 1 • 7^n: 7, 9, 3, 1 • 8^n: 8, 4, 2, 6 Divide n by 4: remainders 1, 2, 3, and 0 correspond to the 1st, 2nd, 3rd, and 4th cycle entries, respectively.

What is the Euclidean division (remainder) formula for positive integers and its remainder constraint?

For an integer dividend N and positive integer divisor D > 0, N = D * Q + R, where Q is the integer quotient and the remainder R is an integer satisfying 0 <= R < D.

What is the formula for the sum of the first n positive integers?

Sum = n(n + 1) / 2. Example: The sum of the first 100 positive integers is 100 * 101 / 2 = 5,050.

What is the divisibility rule for the product of k consecutive integers?

The product of any k consecutive integers is always divisible by k! (k factorial). For example, the product of any 3 consecutive integers is always divisible by 3! = 6.

What prime-factor exponent condition characterizes a positive integer perfect square?

For any integer perfect square N > 1, every prime in its prime factorization has an EVEN exponent (N = p1^(2a) * p2^(2b)...), and its total number of positive divisors is odd. In base 10, an integer perfect square never ends in 2, 3, 7, or 8.

How do you count the number of trailing zeros in n! (n factorial)?

Trailing zeros are formed by 10 = 2 * 5. Since factors of 2 are abundant, count the total factors of 5 in n!: floor(n/5) + floor(n/25) + floor(n/125) + ... Example: In 100!, floor(100/5) + floor(100/25) = 20 + 4 = 24 trailing zeros.

What is the critical difference between (x^a)^b and (x^a) * (x^b)?

• (x^a)^b = x^(a * b) [powers are multiplied] • (x^a) * (x^b) = x^(a + b) [powers are added] Common trap: 2^10 + 2^10 = 2 * (2^10) = 2^11, NOT 2^20 or 4^10.

What is the trap when dividing both sides of an equation by a variable (e.g., x^2 = 5x)?

Never divide by a variable without verifying if the variable could equal 0! Dividing x^2 = 5x by x incorrectly loses the solution x = 0. Instead, factor: x^2 - 5x = 0 => x(x - 5) = 0 => x = 0 or x = 5.

GMAT Focus Quantitative Reasoning: Number Properties & Algebra

32 GMAT Focus Quantitative Reasoning review cards on prime factorization, divisibility, remainders, units-digit cycles, exponents, radicals, absolute value, linear systems, quadratic equations, and inequalities.

Level: basic
Languages: en → en
Cards: 32
Markdown: GMAT Focus Quantitative Reasoning: Number Properties & Algebra (.md)

Flashcards (32)

#	Front (Term / Question)	Back (Definition / Answer)
1	What are the rules for zero and negative exponents?	• For any non-zero x: x^0 = 1 (0^0 is undefined). • x^(-n) = 1 / (x^n). • (a/b)^(-n) = (b/a)^n.
2	What are the common expansions for (a + b)^2 and (a - b)^2, and what common trap must be avoided?	(a + b)^2 = a^2 + 2ab + b^2 (a - b)^2 = a^2 - 2ab + b^2 Trap: (a + b)^2 is NOT equal to a^2 + b^2! The middle cross-term ±2ab is missing.
3	What is the GMAT divisibility rule for 6?	An integer is divisible by 6 if and only if it satisfies both divisibility by 2 (the units digit is even: 0, 2, 4, 6, 8) and divisibility by 3 (the sum of its digits is a multiple of 3).
4	What are Vieta's formulas for the sum and product of quadratic roots?	For ax^2 + bx + c = 0 with roots r1 and r2: • Sum of roots: r1 + r2 = -b / a • Product of roots: r1 * r2 = c / a.
5	What is the GMAT divisibility rule for 3 and 9?	An integer is divisible by 3 if the sum of its digits is divisible by 3. An integer is divisible by 9 if the sum of its digits is divisible by 9.
6	When is the reciprocal inequality rule (if x > y, then 1/x < 1/y) valid?	Only when x and y have the SAME sign (both positive or both negative). If x and y have opposite signs (e.g., x = 2 and y = -3), x > y but 1/x = 1/2 > -1/3 = 1/y (the inequality does NOT flip).
7	What algebraic identity defines the difference of squares and why is it tested so frequently?	a^2 - b^2 = (a - b)(a + b). It appears in disguise, such as x^4 - 16 = (x^2 - 4)(x^2 + 4) = (x - 2)(x + 2)(x^2 + 4), and is frequently used to simplify fractions without long division.
8	What is the parity rule for the product of integers?	Even * Anything = Even; Odd * Odd = Odd. The product of any number of integers is Odd if and only if EVERY factor in the product is Odd.
9	When does a fully reduced fraction a/b produce a terminating decimal?	A fraction in lowest terms terminates if and only if the prime factorization of its denominator b contains NO prime factors other than 2 and/or 5 (e.g., b = 2^m * 5^n). Any other prime factor (3, 7, 11, etc.) results in a repeating decimal.
10	What is the relationship between the GCD, LCM, and two positive integers a and b?	GCD(a, b) * LCM(a, b) = a * b. Note: This product rule holds strictly for two numbers, not necessarily for three or more.
11	What does the quadratic discriminant b^2 - 4ac determine about roots?	For ax^2 + bx + c = 0: • b^2 - 4ac > 0: Two distinct real roots • b^2 - 4ac = 0: Exactly one real root (a double root) • b^2 - 4ac < 0: No real roots (the parabola does not cross the x-axis).
12	When is the sum of n consecutive integers divisible by n?	• If n is ODD, the sum is always a multiple of n (Sum = n * Middle term). • If n is EVEN, the sum is NEVER a multiple of n (Sum = n * (Average of two middle terms), yielding an odd multiple of n/2).
13	How do you factor the sum and difference of cubes?	• Difference of cubes: a^3 - b^3 = (a - b)(a^2 + ab + b^2) • Sum of cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2) Mnemonic 'SOAP': Same sign, Opposite sign, Always Positive for the last term.
14	How do you calculate the total number of positive factors (divisors) for an integer N > 1?	Write the prime factorization of N using distinct primes: N = (p1^a)(p2^b)(p3^c)... Then multiply the incremented exponents: (a + 1)(b + 1)(c + 1)... For example, 72 = 2^3 * 3^2 has (3 + 1)(2 + 1) = 12 positive divisors.
15	When does a linear system a1x + b1y = c1 and a2x + b2y = c2 have infinitely many, zero, or one solution?	Assuming non-zero coefficients a2, b2, c2: • Unique solution (intersecting lines): a1/a2 != b1/b2. • No solution (distinct parallel lines): a1/a2 = b1/b2 != c1/c2. • Infinitely many solutions (coincident lines): a1/a2 = b1/b2 = c1/c2.
16	What are the parity rules for the sum and difference of integers?	Even ± Even = Even; Odd ± Odd = Even; Even ± Odd = Odd. The sum or difference is Odd if and only if the two integers have different parities.
17	What is the formal definition of the absolute value function \|x\| and the principal square root sqrt(x^2)?	\|x\| = x when x >= 0, and \|x\| = -x when x < 0. Crucial GMAT rule: sqrt(x^2) = \|x\|, NOT simply x. If x = -5, sqrt((-5)^2) = sqrt(25) = 5 = -(-5).
18	On the GMAT, does the radical sign sqrt(x) represent positive, negative, or both values?	The radical symbol sqrt(k) ALWAYS represents the non-negative principal square root. sqrt(16) = 4, NOT ±4. Only if an equation says x^2 = 16 do you have two solutions: x = +4 or x = -4.
19	What is the formula for the sum of the first n positive odd integers?	Sum = n^2. Example: 1 + 3 + 5 + 7 = 4^2 = 16. The average of the first n odd positive integers is n.
20	What are the common GMAT traps regarding prime numbers?	1. 2 is the smallest prime number and the ONLY even prime. 2. 1 is NEITHER prime nor composite. 3. Negative integers cannot be prime (primes are positive integers > 1). 4. All primes greater than 3 can be expressed as 6k ± 1 (though not all 6k ± 1 are prime).
21	How do you represent a positive integer N that leaves the same remainder r when divided by both a and b?	If N leaves remainder r when divided by positive integers a and b (with 0 <= r < min(a, b)), then N = LCM(a, b) * k + r for any integer k >= 0. Example: Remainder 2 when divided by 4 and 6 gives N = 12k + 2.
22	What is the GMAT divisibility rule for 4 and 8?	An integer is divisible by 4 if its last two digits form a number divisible by 4 (e.g., 5,324). An integer is divisible by 8 if its last three digits form a number divisible by 8 (e.g., 17,816).
23	For what values of x is x^2 > x?	x^2 > x is true when x > 1 OR x < 0. It is FALSE when 0 <= x <= 1 (e.g., (1/2)^2 = 1/4 < 1/2; 0^2 = 0; 1^2 = 1). Testing the (0, 1) boundary is high-yield.
24	How are fractional exponents x^(a/b) defined in real-number algebra?	For a positive real base x > 0 and integers a, b with b > 0, x^(a/b) = (b-th root of x)^a = b-th root of (x^a) (and for negative real x when b is odd and a/b is reduced). Example: 8^(2/3) = (cube root of 8)^2 = 2^2 = 4.
25	What is the units-digit cycle pattern for positive integer powers of 2, 3, 7, and 8?	All four have a repeating units-digit period of 4 for positive integer exponents n >= 1: • 2^n: 2, 4, 8, 6 • 3^n: 3, 9, 7, 1 • 7^n: 7, 9, 3, 1 • 8^n: 8, 4, 2, 6 Divide n by 4: remainders 1, 2, 3, and 0 correspond to the 1st, 2nd, 3rd, and 4th cycle entries, respectively.
26	What is the Euclidean division (remainder) formula for positive integers and its remainder constraint?	For an integer dividend N and positive integer divisor D > 0, N = D * Q + R, where Q is the integer quotient and the remainder R is an integer satisfying 0 <= R < D.
27	What is the formula for the sum of the first n positive integers?	Sum = n(n + 1) / 2. Example: The sum of the first 100 positive integers is 100 * 101 / 2 = 5,050.
28	What is the divisibility rule for the product of k consecutive integers?	The product of any k consecutive integers is always divisible by k! (k factorial). For example, the product of any 3 consecutive integers is always divisible by 3! = 6.
29	What prime-factor exponent condition characterizes a positive integer perfect square?	For any integer perfect square N > 1, every prime in its prime factorization has an EVEN exponent (N = p1^(2a) * p2^(2b)...), and its total number of positive divisors is odd. In base 10, an integer perfect square never ends in 2, 3, 7, or 8.
30	How do you count the number of trailing zeros in n! (n factorial)?	Trailing zeros are formed by 10 = 2 * 5. Since factors of 2 are abundant, count the total factors of 5 in n!: floor(n/5) + floor(n/25) + floor(n/125) + ... Example: In 100!, floor(100/5) + floor(100/25) = 20 + 4 = 24 trailing zeros.
31	What is the critical difference between (x^a)^b and (x^a) * (x^b)?	• (x^a)^b = x^(a * b) [powers are multiplied] • (x^a) * (x^b) = x^(a + b) [powers are added] Common trap: 2^10 + 2^10 = 2 * (2^10) = 2^11, NOT 2^20 or 4^10.
32	What is the trap when dividing both sides of an equation by a variable (e.g., x^2 = 5x)?	Never divide by a variable without verifying if the variable could equal 0! Dividing x^2 = 5x by x incorrectly loses the solution x = 0. Instead, factor: x^2 - 5x = 0 => x(x - 5) = 0 => x = 0 or x = 5.