6.1 Integers, Divisibility, Primes, Remainders & Odd/Even

Integers, Divisibility, Primes, Remainders & Odd/Even

Quick Answer: GRE arithmetic starts with integers: divisibility rules let you test factors in seconds, prime factorization is the backbone of GCD and LCM, and odd/even parity rules decide whether a sum or product is even without computation. Master these and most "number theory" Quantitative Comparison questions become 20-second problems.

The integers are the set {..., −3, −2, −1, 0, 1, 2, 3, ...}. The integer 0 is neither positive nor negative. A positive integer n has divisors d where n ÷ d leaves remainder 0; we say "d divides n" (written d | n).

Divisibility Rules (Memorize These)

The GRE rewards fast divisibility testing. Use this table for any integer up to a few thousand:

A number divisible by 12 must satisfy the rules for 3 and 4 (since 12 = 3 × 4 and gcd(3, 4) = 1). For 8, look at the last three digits: 5,648 is divisible by 8 because 648 ÷ 8 = 81.

Worked Example — Divisibility Combo

Is 4,356 divisible by 6?

• Divisible by 2? Last digit 6, yes.
• Divisible by 3? Digit sum 4 + 3 + 5 + 6 = 18, divisible by 3, yes.
• Both conditions hold, so 4,356 is divisible by 6. (Check: 4,356 ÷ 6 = 726.)

Prime Numbers and the Fundamental Theorem of Arithmetic

A prime is an integer greater than 1 whose only positive divisors are 1 and itself. The first ten primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. The number 1 is not prime; 2 is the only even prime.

The Fundamental Theorem of Arithmetic says every integer n > 1 has a unique prime factorization (up to ordering). To factor, divide by the smallest prime repeatedly: 72 = 2 × 36 = 2 × 2 × 18 = 2 × 2 × 2 × 9 = 2 × 2 × 2 × 3 × 3 = 2³ × 3².

This is your single most useful technique for GCD/LCM and for simplifying fractions with large numerators. When a QC question involves two large numbers, factor both — the comparison usually collapses once shared prime factors cancel.

GCD, LCM, and the Identity That Ties Them

The greatest common divisor GCD(a, b) is the largest integer dividing both. The least common multiple LCM(a, b) is the smallest positive integer divisible by both. The cleanest way to compute both is via prime factorization:

• GCD: take the minimum exponent for each shared prime.
• LCM: take the maximum exponent for each prime that appears.

For a = 72 = 2³ × 3² and b = 60 = 2² × 3 × 5:

• GCD = 2² × 3¹ = 12.
• LCM = 2³ × 3² × 5 = 360.

The identity GCD(a, b) × LCM(a, b) = a × b lets you check or compute one from the other: 12 × 360 = 4,320 = 72 × 60. ✓

Worked Example — LCM Scheduling

Bus A leaves every 8 minutes; Bus B every 12 minutes. They leave together at 8:00. When next together? LCM(8, 12) = 24, so 8:24.

Remainders and Modular Thinking

If a ÷ b gives quotient q and remainder r, then a = bq + r with 0 ≤ r < b. GRE often tests remainders indirectly: "What is the units digit of 7^42?" Powers cycle through last digits. 7¹ = 7, 7² = 49 → 9, 7³ → 3, 7⁴ → 1, then repeats every 4. So 7^42 = 7^(4·10+2), same last digit as 7² → 9.

Modular arithmetic shorthand: "a ≡ r (mod b)" means a and r leave the same remainder when divided by b. Useful for QC: if x ≡ 3 (mod 5), x could be 3, 8, 13, 18, ..., so x is NOT necessarily equal to 3 — eliminate (C) and check whether Quantity B lies in the residue class.

Odd/Even Parity Rules

Odd integers end in 1, 3, 5, 7, 9; even integers end in 0, 2, 4, 6, 8. Parity is preserved under addition and multiplication in fixed patterns:

A sum of two integers is odd only if one is odd and the other even. A product is odd only if every factor is odd. Use this to QC-test variable expressions: if Quantity A is (2k + 1)(2m) for integers k, m, it is even — no arithmetic needed.

Worked Example — Parity Saves Time

If n is an integer and n² is even, is n³ + n divisible by 4?

• n² even ⟹ n even (only even × even = even).
• So n = 2k. Then n³ + n = 8k³ + 2k = 2k(4k² + 1).
• If k is even, 2k is divisible by 4 ✓.
• If k is odd, 4k² + 1 = 4(odd) + 1 = odd, and 2k has exactly one factor of 2, so the product has only one factor of 2 — NOT divisible by 4.
• Answer: not always. Counterexample: k = 1, n = 2: n³ + n = 8 + 2 = 10, not divisible by 4.

That is the kind of "is it always true?" reasoning GRE Quantitative Comparison loves. When a statement depends on a parameter, test one even and one odd value; if the answer flips, choose (D).