Prime Factorization, Divisors, and LCM/GCD
Number theory is the area of GMAT Quantitative that most directly rewards conceptual understanding over mechanical computation. Questions testing divisibility, prime factorization, and LCM/GCD appear consistently in the 700+ range and can be solved quickly by students who understand the underlying principles — but are extremely time-consuming for those who test specific values without theoretical insight.
Prime factorization divisor counting: every positive integer has a unique prime factorization (the Fundamental Theorem of Arithmetic). For n = p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ, the total number of positive divisors = (a₁ + 1)(a₂ + 1)...(aₖ + 1). For example, 360 = 2³ × 3² × 5¹, so it has (3+1)(2+1)(1+1) = 4 × 3 × 2 = 24 divisors. This formula allows questions like 'If n has exactly 6 positive divisors, what are the possible forms of n?' to be answered systematically: 6 = 6×1 (n = p⁵) or 6 = 3×2 (n = p² × q, where p and q are distinct primes).
LCM and GCD with prime factorizations: GCD takes the minimum exponent for each prime across both numbers; LCM takes the maximum exponent. For GCD(2³ × 3² × 5, 2² × 3³ × 7): GCD = 2² × 3² = 36; LCM = 2³ × 3³ × 5 × 7 = 7560. The relationship LCM(a,b) × GCD(a,b) = a × b is frequently useful on GMAT. 'If LCM(m,n) = 120 and GCD(m,n) = 4, what is mn?' → mn = 120 × 4 = 480.
For questions about the form of n given divisibility constraints: if a question states 'n is divisible by both 4 and 6', this means n contains 2² (from 4 = 2²) and 2 × 3 (from 6 = 2 × 3). The combined requirement is 2² × 3 — not 2 × 3 (which would be GCD of 4 and 6 = 2) and not 2³ × 3 (which would be LCM = 12 but unnecessarily high). The minimum n satisfying both constraints is LCM(4, 6) = 12. This LCM interpretation of 'divisible by both X and Y' applies directly to many GMAT word problems.