Number Properties
Primes, factors, divisibility, remainders
Key Concepts
9 conceptsPrime Numbers
A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself. The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Note that 2 is the only even prime. The GMAT frequently tests whether you can quickly identify primes and apply prime factorization to solve problems involving LCM, GCD, and divisibility.
Prime Factorization
Every positive integer greater than 1 can be expressed as a unique product of prime numbers. For example, 360 = 2^3 x 3^2 x 5. This decomposition is the key to solving problems about factors, multiples, LCM, and GCD. To find the number of factors of any integer, add 1 to each exponent in its prime factorization and multiply. For 360: (3+1)(2+1)(1+1) = 24 factors.
Divisibility Rules
Quick tests save time on the GMAT. Divisible by 2: last digit is even. By 3: digit sum divisible by 3. By 4: last two digits form a number divisible by 4. By 5: ends in 0 or 5. By 6: divisible by both 2 and 3. By 8: last three digits divisible by 8. By 9: digit sum divisible by 9. By 11: alternating sum of digits is divisible by 11.
GCD and LCM
The Greatest Common Divisor (GCD) of two numbers is the largest number that divides both evenly. The Least Common Multiple (LCM) is the smallest number that both divide into evenly. Key relationship: GCD(a,b) x LCM(a,b) = a x b. To find GCD, take the minimum power of each shared prime factor. To find LCM, take the maximum power of each prime factor that appears in either number.
Remainders and Modular Arithmetic
The remainder when a is divided by b is the leftover value after removing the largest possible multiple of b. Remainders follow useful arithmetic: if a leaves remainder r1 when divided by n, and b leaves remainder r2, then (a+b) leaves the same remainder as (r1+r2) when divided by n, and (a x b) leaves the same remainder as (r1 x r2) when divided by n. This lets you simplify huge numbers before dividing.
Odd and Even Integer Rules
Even plus even is even; odd plus odd is even; even plus odd is odd. Even times anything is even; odd times odd is odd. These parity rules let you quickly rule out answer choices in Data Sufficiency questions about whether an unknown integer is odd or even, without needing the integer's actual value.
Perfect Squares and Perfect Cubes
A perfect square is an integer that is the square of another integer (1, 4, 9, 16, 25...); its prime factorization always has every exponent even. A perfect cube (1, 8, 27, 64...) has every exponent in its prime factorization divisible by 3. These properties are the basis for GMAT questions asking what must be multiplied or divided to make a number a perfect square or cube.
Factorials and Trailing Zeros
A factorial n! is the product of all positive integers up to n. The number of trailing zeros in n! equals the number of times 10 (i.e., pairs of 2 and 5) divides into n!, which is determined by counting factors of 5 in the numbers from 1 to n, since factors of 2 are always more plentiful than factors of 5.
Consecutive Integer Properties
Among any set of n consecutive integers, exactly one is divisible by n, and the product of n consecutive integers is always divisible by n!. Sums of consecutive integers also follow patterns: the sum of an odd count of consecutive integers is always divisible by the count, since the middle term equals the average.