Number Properties
Primes, factors, divisibility, remainders
Key Concepts
4 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.