Number properties: complete GMAT guide

Integers include positive numbers, negative numbers, and zero. Zero is even, neither positive nor negative, and divisible by every nonzero integer. A question that says "integer" allows negative values and zero unless it says "positive integer" or "nonnegative integer."

Always check the allowed set before testing cases. If $x$ is an integer, then $x$ can be negative. If $x$ is a positive integer, then $x \ge 1$. These differences decide many Data Sufficiency questions.

A prime number has exactly two positive factors: 1 and itself. The primes start 2, 3, 5, 7, 11, 13, 17, 19, 23, and so on. The number 1 is not prime because it has only one positive factor. The number 2 is the only even prime.

A composite number has more than two positive factors. Every composite number can be written as a product of primes, and that factorization is unique apart from order. This fact is the basis for questions about factors and divisibility.

Parity rules are simple but easy to misapply:

A common question asks whether an expression is even or odd. Test the possible parities of the variables. For example, $n^2+n=n(n+1)$ is always even because one of two consecutive integers is even. This does not depend on whether $n$ is positive or negative.

The product or quotient of two numbers with the same sign is positive. The product or quotient of two numbers with opposite signs is negative. A product is negative only when an odd number of negative factors are present.

This rule matters in inequality questions. If $xy \gt 0$, then $x$ and $y$ have the same sign. If $xy \lt 0$, they have opposite signs. If $x^2 \gt 0$, then $x \neq 0$, but $x$ can be positive or negative.

Consecutive integers differ by 1. If the smallest is $n$, the next are $n+1$, $n+2$, and so on. Consecutive even or odd integers differ by 2, so they can be written as $2n$, $2n+2$, $2n+4$, or as $2n+1$, $2n+3$, $2n+5$.

Among any $k$ consecutive integers, exactly one is divisible by $k$. Among any two consecutive integers, one is even. Among any three consecutive integers, one is divisible by 3. These facts answer many "always divisible" questions without algebra.

The average of an odd number of consecutive integers is the middle integer. For three consecutive integers, the middle one is the average. For five, the third one is the average. This shortcut saves time when the sum is given.

A factor of $n$ divides $n$ without a remainder. A multiple of $n$ is the product of $n$ and an integer. Every positive integer is both a factor and a multiple of itself.

Factors come in pairs. For 36, the pairs are $(1,36)$, $(2,18)$, $(3,12)$, $(4,9)$, and $(6,6)$. The square root appears once, which is why a perfect square has an odd number of factors.

The parity of a remainder depends on the divisor. When dividing by 2, the remainder is either 0 or 1. When dividing by 3, it is 0, 1, or 2. A number that leaves remainder 2 when divided by 5 can be written as $5k+2$.

Do not assume that a remainder of 2 means the number is even. For example, 17 leaves remainder 2 when divided by 5, and 17 is odd. Remainders describe position relative to the divisor, not parity.

Number properties questions in Data Sufficiency often hinge on whether a variable can be negative or fractional. Test these cases:

• positive integer
• negative integer
• zero
• positive fraction between 0 and 1
• negative fraction between -1 and 0

A statement that is sufficient for positive integers may fail for fractions. For example, if $x \gt 0$, then $x^2 \gt x$ only when $x \gt 1$. For $0 \lt x \lt 1$, $x^2 \lt x$. This is why the allowed set matters.

Remainder questions often repeat in a cycle. If a number leaves remainder 2 when divided by 5, its possible values are 2, 7, 12, 17, and so on. Adding or multiplying such numbers produces a pattern in the remainders. You can work with the remainders directly instead of the full numbers.

For example, if $a$ leaves remainder 3 when divided by 7 and $b$ leaves remainder 4, then $a+b$ leaves remainder 7, which is equivalent to remainder 0. The product $ab$ leaves remainder $12$, which is equivalent to remainder 5. This modular arithmetic is a fast way to handle large numbers.

1. Identify the allowed values: integers, positives, nonnegatives, or reals.
2. Test parity and sign separately.
3. Use prime factorization for factor questions.
4. Use consecutive integer facts for "always divisible" questions.
5. Test zero, negatives, and fractions when the question allows them.
6. Check whether the statement actually determines the target.

• Treating 1 as a prime.
• Forgetting that 2 is prime.
• Assuming a variable is positive because it looks positive.
• Confusing factors with multiples.
• Assuming a perfect square always has an even number of factors.
• Testing only integers when fractions are allowed.
• Using parity rules on expressions with subtraction without checking signs.