Integers
Numbers, such as -1, 0, 1, 2, and 3, that have no fractional part. Integers include the counting numbers (1, 2, 3, …), their negative counterparts (-1, -2, -3, …), and 0.
Divisibility
If an integer x divided by another number y yields an integer, then the x is said to be divisible by y.
Example:
12 divided by 3 yields the integer 4. Therefore, 12 is divisible by 3.
12 divided by 5 does not yield an integer. Therefore, 12 is not divisible by 5.
Factors
Positive integers that divide evenly into an integer. Factors are equal to or smaller than the integer in question. 12 is a factor of 12, as are 1, 2, 3, 4, and 6.
Multiples
Multiples are integers formed by multiplying some integer by any other integer. 12 is a multiple of 12 (12 * 1), as are 24 (=12 * 2), 36 (=12 * 3), 48 (=12 * 4), and 60 (=12 * 5). (Negative multiples are also possible in mathematics but are not typically tested on the GMAT. Likewise, 0 is technically a multiple of every integer, since 0 * that integer = 0, which is itself an integer; however, this fact is also generally not tested on the GMAT, which usually restricts divisibility questions to positive integers. Therefore, on the GMAT, it is helpful to think of multiples as equal to or larger than the integer in question.)
Divisibility & Multiples
If you add two non-multiples of N, the result could be either a multiple of N or a non-multiple of N. We can’t tell without checking the specific numbers!
Example:
If you add or subtract two (or more) multiples of N, the result will also be a multiple of N. 16 + 12 = 28. (Multiple of 4) + (Multiple of 4) = (Multiple of 4)
If you add (or subtract) a multiple of N to (from) a non-multiple of N, the result is a non-multiple of N. 16 – 6 = 10. (Multiple of 4) – (Non-Multiple of 4) = (Non-Multiple of 4)
Primes
A positive integer with exactly two factors: 1 and itself. The number 1 does not qualify as prime because it has only one factor, not two. The number 2 is the smallest prime number; it is also the only even prime number. The numbers 2, 3, 5, 7, 11, 13 etc. are prime.
Prime Factorization
Prime factorization is a way to express any number as a product of prime numbers. For example, the prime factorization of 60 is 2 * 2 * 3 * 5. Prime factorization is useful in answering questions about divisibility.
Factor Pairs
Factor pairs for any integer are the pairs of factors that, when multiplied together, yield the integer. For example, the factor pairs of 60 are 1 & 60, 2 & 30, 3 & 20, 4 & 15, 5 & 12, and 6 & 10.
Factor Foundation Rule
If a is a factor of b, and b is a factor of c, then a is also a factor of c. For example, 2 is a factor of 10. 10 is a factor of 60. Therefore, 2 is also a factor of 60.
Greatest Common Factor
Greatest Common FACTOR refers to the largest factor of two (or more) integers. Factors will be equal to or smaller than the starting integers. The GCF of 12 and 30 is 6 because 6 is the largest number that goes into both 12 and 30.
Least Common Multiple
Least Common MULTIPLE refers to the smallest multiple of two (or more) integers. Multiples will be equal to or larger than the starting integers. The LCM of 12 and 30 is 60 because 60 is the smallest number that both 12 and 30 go into.
Note:
Both of these terms ask you to find some number relative to a set of two or more other numbers.
Factorial
The factorial symbol is an exclamation mark (!) and is defined as the product of all integers from 1 up to and including the given number. 5! = 5 * 4 * 3 * 2 * 1 = 120.
Remainder
The integer portion of the dividend (or numerator, when written as a fraction) that is not evenly divisible by the divisor (or denominator). (10/3) or 10 ÷ 3 = 3 with a remainder of 1. 3 goes into 10 three times, for a total of 3 * 3 = 9, and 1 is left over. The 1 is not evenly divisible by the divisor, 3, so that 1 is called the remainder.
Dividend
The numerator of a fraction, or the part of a division operation that comes before the division sign. In the operation 22 ÷ 4, 22 is the dividend.
Divisor
The denominator of a fraction, or the part of a division operation that comes after the division sign. In the operation 22 / 4, 4 is the divisor.
Quotient
Part of the result of a division operation; the quotient represents the part of the result that is evenly divisible. In the operation 22 ÷ 4, the quotient is 5, because 4 goes into 22 evenly a total of five times (4*5 = 20). The left over portion, in this case 2, is the remainder.
