An integer is any number in the set {… −3, −2, −1, 0, 1, 2, 3, …}. For any integer n, the numbers n, n+1, n+2, n+3, … are consecutive integers. Zero is neither positive nor negative, and it is even.

A) Divisors, Factors, and Multiples

For integers x and y, if x ≠ 0, then x is a divisor or factor of y if y = xn for some integer n. Then y is divisible by x and is a multiple of x.

• Since 28 = (7)(4), both 4 and 7 are divisors of 28. But 8 is not a divisor of 28, since 28/8 = 3.5 — not an integer.
• Factor: factors of 12 are 1, 2, 3, 4, 6, 12.
• Multiple: multiples of 4 are 4, 8, 12, 16, …

B) Even and Odd

• Even: divisible by 2. Set: {… −4, −2, 0, 2, 4, 6, 8, …}. For any integer n, consecutive even integers are 2n, 2n+2, 2n+4, …
• Odd: not divisible by 2. Set: {… −3, −1, 1, 3, 5, …}. Consecutive odd integers: 2n+1, 2n+3, 2n+5, …
• Even × anything = even. Even + even = even. Even + odd = odd. Odd × odd = odd. Odd + odd = even.
• Among any n consecutive integers, exactly one is divisible by n.
• The product of n consecutive integers is divisible by n!

C) Divisibility Rules

• 2: last digit is even (0, 2, 4, 6, 8)
• 3: sum of digits is divisible by 3 (e.g., 123 → 1+2+3=6 ✓)
• 4: last two digits form a number divisible by 4 (e.g., 1,312 → 12÷4=3 ✓)
• 5: last digit is 0 or 5
• 6: divisible by both 2 and 3
• 8: last three digits form a number divisible by 8
• 9: sum of digits is divisible by 9 (e.g., 729 → 7+2+9=18 ✓)
• 10: last digit is 0
• 11: alternating sum of digits is divisible by 11 (e.g., 121 → 1−2+1=0 ✓; 253 → 2−5+3=0 ✓)
• 25: last two digits are 00, 25, 50, or 75

D) Remainders

When a ÷ b = q remainder r: then a = bq + r, where 0 ≤ r < b. Example: 28 ÷ 8 → quotient 3, remainder 4, because 28 = (8)(3) + 4.

• Remainder when dividing by 2 tells you odd/even (remainder 0 = even, remainder 1 = odd).
• If two numbers have the same remainder when divided by n, their difference is divisible by n.
• Smaller ÷ larger: quotient = 0, remainder = the smaller number. Example: 3 ÷ 7 → quotient 0, remainder 3.
• Last-digit trick: remainder when dividing by 10 = last digit of the dividend.
• Remainder of a sum: to find the remainder of (a + b) ÷ n, find each remainder separately, add them, then take that remainder. ("mod" just means "the remainder when divided by" — so 17 mod 5 = 2, because 17 ÷ 5 leaves remainder 2.) Example: (17 + 14) mod 5 → 17 mod 5 = 2, 14 mod 5 = 4, (2+4) mod 5 = 6 mod 5 = 1.
• Two-constraint problems (LCM method): find x that satisfies two remainder conditions simultaneously. Write out values satisfying condition 1, then check each against condition 2. The pattern repeats every LCM of the two divisors.
  Example: find smallest positive x where (x ÷ 5 leaves remainder 2) AND (x ÷ 7 leaves remainder 3).
  Values with x mod 5 = 2: 2, 7, 12, 17, 22, 27, 32, …
  Check x mod 7 = 3: 2÷7 rem 2 ✗ · 7÷7 rem 0 ✗ · 12÷7 rem 5 ✗ · 17÷7 rem 3 ✓ → x = 17.
  Next solution = 17 + LCM(5,7) = 17 + 35 = 52. Every 35 after that also works.

E) Consecutive Integers — Sum Divisibility

• Sum of n consecutive integers: if n is odd, the sum is always divisible by n (the middle number is the mean, and sum = n × mean). Example: 3 + 4 + 5 = 12, and 12 ÷ 3 = 4 ✓. Another: 1 + 2 + 3 + 4 + 5 = 15, and 15 ÷ 5 = 3 ✓.
• If n is even, the sum is never divisible by n (the mean is a half-integer). Example: 3 + 4 + 5 + 6 = 18, and 18 ÷ 4 = 4.5 — not an integer ✓.

F) Sum Formulas for Integer Series

These shortcuts avoid adding every number one by one.

• Sum of first n positive integers: 1+2+…+n = n(n+1)/2. Example: 1+2+…+100 = 100×101/2 = 5050.
• Sum of first n odd integers: 1+3+5+…+(2n−1) = n². Example: first 5 odds (1+3+5+7+9) = 5² = 25.
• Sum of first n even integers: 2+4+6+…+2n = n(n+1). Example: first 4 evens (2+4+6+8) = 4×5 = 20.

G) Prime Numbers

A prime has exactly two factors: 1 and itself. 1 is not prime. 2 is the only even prime. Every integer greater than 1 is either prime or a unique product of prime factors (Fundamental Theorem of Arithmetic).

Key primes to memorize through 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.

• All primes > 2 end in 1, 3, 7, or 9 (no even digits, not ending in 5).
• All primes > 3 are of the form 6n − 1 or 6n + 1. This means every prime > 3 is adjacent to a multiple of 6. Useful for testing primality: check if n±1 is a multiple of 6.
• Between any integer n and 2n (n > 1), there is always at least one prime (Bertrand's postulate). Useful in DS constraints.

H) Perfect Square Properties

A perfect square is an integer that equals some integer squared (1, 4, 9, 16, 25, …).

• A perfect square always has an odd number of distinct factors (because one factor pairs with itself at the square root).
• In the prime factorization of a perfect square, all prime factors have even exponents. Example: 36 = 2² × 3². Conversely, if any prime exponent is odd, the number is not a perfect square.
• The square root of a perfect square is an integer; the square root of any other positive integer is irrational.

I) GCD and LCM

• GCD (Greatest Common Divisor): use prime factorization — take the minimum power of each shared prime. GCD(12, 18): 12 = 2²×3, 18 = 2×3² → GCD = 2×3 = 6.
• LCM (Least Common Multiple): take the maximum power of each prime across both numbers. LCM(12, 18) = 2²×3² = 36.
• Key relationship: GCD(a,b) × LCM(a,b) = a × b (for any two positive integers).

J) Number of Factors Formula

If N = p^a × q^b × r^c (prime factorization), then the total number of factors = (a+1)(b+1)(c+1)…

Example: 12 = 2² × 3¹ → factors = (2+1)(1+1) = 6. Factors of 12: 1, 2, 3, 4, 6, 12 ✓

• Sum of all factors of N = [(p^a+1−1)/(p−1)] × [(q^b+1−1)/(q−1)] × …
• Product of all factors = N^d/2 where d = number of factors.

K) Evenly-Spaced Sets

An evenly-spaced set has a constant gap between every term: {10, 15, 20, 25, …} or {3, 7, 11, 15, …}. Consecutive integers are a special case (gap = 1).

• Count of terms: (last − first) / spacing + 1.
  Example: integers from 10 to 40 divisible by 5 → (40−10)/5 + 1 = 6 + 1 = 7 (verify: 10, 15, 20, 25, 30, 35, 40 ✓).
• Sum: count × (first + last) / 2.
  Example: 10+15+20+25+30+35+40 = 7 × (10+40)/2 = 7 × 25 = 175.
• Mean = Median: for any evenly-spaced set, the mean equals the median — both equal the middle term (or average of the two middle values for even count). This also equals (first + last) / 2.
• Consecutive even integers: spacing 2, e.g., {2, 4, 6, 8}. Consecutive odd integers: spacing 2, e.g., {1, 3, 5, 7}.

L) Trailing Zeros in n!

Trailing zeros come from pairs of 2 × 5 in a factorial. Since 2s are always more plentiful than 5s, just count factors of 5. The floor symbol ⌊ ⌋ means "divide and drop any decimal — round down." So ⌊47/5⌋ = ⌊9.4⌋ = 9.

Formula: trailing zeros = ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + …

Example: 100! trailing zeros = ⌊100/5⌋ + ⌊100/25⌋ + ⌊100/125⌋ = 20 + 4 + 0 = 24.

M) Exponents and Roots

An exponent tells you how many times to multiply a number by itself. 2³ = 2 × 2 × 2 = 8. The base is 2, the exponent is 3.

• x^a × x^b = x^a+b   (same base: add exponents)   Numeric: 3² × 3⁴ = 3⁶ = 729   Algebraic: x³ × x⁵ = x⁸
• x^a ÷ x^b = x^a−b   (same base: subtract exponents)   Numeric: 5⁵ ÷ 5² = 5³ = 125   Algebraic: x⁷ ÷ x² = x⁵
• (x^a)^b = x^ab   (power of a power: multiply exponents)   Numeric: (2³)⁴ = 2¹²   Algebraic: (x²)³ = x⁶
• (xy)^a = x^a × y^a   Numeric: (2×3)⁴ = 2⁴ × 3⁴ = 1296   Algebraic: (2x)³ = 8x³
• x⁰ = 1 for any x ≠ 0   (7⁰ = 1, 100⁰ = 1)
• x^−a = 1/x^a   (negative exponent = flip to denominator)   Example: 2⁻³ = 1/8
• x^1/n = ⁿ√x   (fractional exponent = nth root)   Example: 8^1/3 = ∛8 = 2
• Squaring a number > 1 makes it larger. Squaring a number between 0 and 1 makes it smaller (e.g., 0.5² = 0.25). Squaring any negative makes it positive.
• √9 = 3 by convention (the positive root only). But x² = 9 → x = 3 or x = −3.

Radical rules:

• √(ab) = √a × √b   Example: √12 = √4 × √3 = 2√3
• √(a/b) = √a / √b   Example: √(9/16) = 3/4
• √(a²) = |a|   (always the positive result — √(x²) = |x|, not just x)
• Rationalizing a denominator: eliminate the root from the denominator by multiplying top and bottom by √b.   Example: 5/√3 × (√3/√3) = 5√3/3
• Conjugate rationalization: if the denominator is (a + √b), multiply by (a − √b).   Example: 1/(2+√3) × (2−√3)/(2−√3) = (2−√3)/(4−3) = 2−√3

N) Cube Roots & nth Roots

A cube root asks: "what number, multiplied by itself 3 times, gives this result?" ∛8 = 2 because 2 × 2 × 2 = 8. ∛27 = 3 because 3³ = 27. Unlike square roots, cube roots work on negatives: ∛(−8) = −2 because (−2)³ = −8.

• General nth root: x^1/n = ⁿ√x. Even roots (square, 4th, 6th…) require x ≥ 0 — you can't take an even root of a negative. Odd roots (cube, 5th…) work for any real number.

O) Unit Digit Cycles

• 2: 2, 4, 8, 6 (cycle 4)
• 3: 3, 9, 7, 1 (cycle 4)
• 7: 7, 9, 3, 1 (cycle 4)
• 8: 8, 4, 2, 6 (cycle 4)
• 4: 4, 6 (cycle 2)   9: 9, 1 (cycle 2)
• 0, 1, 5, 6: always end in themselves

To find unit digit of xⁿ: find n mod (cycle length), use that position. Example: unit digit of 7⁵³: 53 mod 4 = 1 → same as 7¹ → unit digit 7.

P) Properties of Arithmetic Operations

The GMAT Official Guide presents these foundational rules as a reference for all arithmetic. They underlie every algebraic manipulation — knowing them cold prevents errors and speeds up simplification.

Addition and Subtraction

• x + 0 = x and x − 0 = x   (adding or subtracting zero changes nothing)
• x − x = 0   (a number minus itself is zero)
• x + y = y + x   (Commutative property of addition: order doesn't matter)
• x − y = −(y − x) = x + (−y)   (subtraction as adding the negative)
• (x + y) + z = x + (y + z)   (Associative property of addition: grouping doesn't matter)
• If x and y are both positive, then x + y is positive.
• If x and y are both negative, then x + y is negative.

Multiplication and Division

• x × 1 = x   (multiplying by 1 changes nothing)
• x × 0 = 0   (multiplying by 0 always gives 0)
• If x ≠ 0, then x/x = 1   (a nonzero number divided by itself is 1)
• 0/x is undefined when x = 0   (never divide by zero)
• xy = yx   (Commutative property of multiplication: order doesn't matter)
• (xy)z = x(yz)   (Associative property of multiplication: grouping doesn't matter)
• xy + xz = x(y + z)   (Distributive property: factor out a common term)
• If x ≠ 0 and y ≠ 0, then x/y = 1/(y/x)   (dividing by y = multiplying by 1/y)
• If x and y are both positive, then xy is positive.
• If x is positive and y is negative (or vice versa), then xy is negative.
• If x and y are both negative, then xy is positive (negative × negative = positive).
• If xy = 0, then x = 0 or y = 0, or both. (Zero product property)

Exponentiation

• x¹ = x   (any number to the first power equals itself)
• x⁰ = 1 for any x ≠ 0   (any nonzero number to the zero power equals 1)
• If x ≠ 0, then x⁻¹ = 1/x   (negative exponent = reciprocal)
• (x^y)^z = x^yz   (power of a power: multiply exponents)
• x^y+z = x^y · x^z   (same base, different exponents: add exponents when multiplying)
• If x ≠ 0, then x^y−z = x^y/x^z   (same base: subtract exponents when dividing)
• (xz)^y = x^yz^y   (power distributes over multiplication)
• If z ≠ 0, then (x/z)^y = x^y/z^y   (power distributes over division)
• If z ≠ 0, then x^y/z = (x^1/z)^y   (fractional exponent = root)

Sign Rules — Quick Reference

Distributive Property — How to Use It

The distributive property a(b + c) = ab + ac is the most frequently applied property in algebra. It works in both directions:

• Expand: 3(x + 4) = 3x + 12   (distribute 3 to each term inside)
• Factor: 6x + 9 = 3(2x + 3)   (pull out the common factor 3)
• With subtraction: a(b − c) = ab − ac. Example: 5(x − 2) = 5x − 10.
• Negative outside: −(a + b) = −a − b. Example: −(3x + 4) = −3x − 4. A common error is forgetting to distribute the negative to every term.

Operation Property Traps

• 0/x = 0 (zero divided by anything nonzero is zero), but x/0 is undefined. These are different.
• x⁰ = 1 for any x ≠ 0, but 0⁰ is indeterminate — the GMAT won't test this edge case.
• (−x)² = x² ≥ 0, but −x² ≤ 0. The parentheses matter: (−3)² = 9 but −3² = −9.
• Negative × negative = positive. It's common to lose track of signs when a chain of negatives is involved. Count the negatives: even count = positive result, odd count = negative result.
• Distributive trap: (x + y)² ≠ x² + y². You must expand: (x + y)² = x² + 2xy + y².