Data Sufficiency: arithmetic

A value question asks for a specific number, such as "What is the remainder when $n$ is divided by 7?" A statement is sufficient only if it produces exactly one value. A yes/no question asks something like "Is $n$ divisible by 6?" A statement is sufficient only if it forces the same answer for every allowed value.

These standards differ. A statement that narrows the possibilities to two values is not sufficient for a value question, even if both values are close. A statement that always produces "yes" is sufficient for a yes/no question, even if it does not reveal the value.

Test values that stress the conditions: positive, negative, zero, fractions, and numbers near boundaries. If two allowed cases produce different answers, the statement is not sufficient. If you cannot find a counterexample after a reasonable set, the statement is likely sufficient, but you should also look for algebraic reasoning.

Remainder statements often use the form $n=divisor \times quotient+remainder$. If a statement says $n$ leaves remainder 3 when divided by 5, write $n=5k+3$. Then test expressions such as $n+2$ or $2n$ by substitution.

The combined case works because the conditions force $n$ to be congruent to 2 modulo 6. Testing a few values reveals the pattern, and the pattern is the proof.

A statement that says a number is divisible by 4 and by 6 does not guarantee divisibility by 24, because 4 and 6 share a factor of 2. The number 12 is a counterexample. To guarantee divisibility by 24, the statement must provide enough independent prime factors: $24=2^3 \times 3$.

For factor questions, prime factorization is often the fastest way to test a statement. If a statement says $n$ is divisible by 12, then $n$ contains $2^2 \times 3$. If the question asks whether $n$ is divisible by 8, the statement is not sufficient, because $n$ might have only $2^2$.

Parity statements are common in yes/no questions. If a statement says $n$ is odd, then $n+1$ is even and $n^2$ is odd. If a statement says $n$ is even, then $n^2$ is divisible by 4, because $n=2k$ implies $n^2=4k^2$.

Test the interaction of two variables. If $x$ is even and $y$ is odd, then $x+y$ is odd and $xy$ is even. If a statement gives only $x+y$ is odd, then $x$ and $y$ have opposite parity, but you do not know which is which.

Prime statements often hinge on the number 2. If a statement says $p$ is a prime greater than 2, then $p$ is odd. If it says $p$ is prime, then $p$ could be 2. The difference can decide a yes/no question about parity.

Be careful with negative numbers. A prime is defined as a positive integer greater than 1 with exactly two positive factors. Negative integers are not prime on the GMAT.

Statements about the greatest common factor or least common multiple can determine a pair of numbers when combined with other information. For two positive integers $a$ and $b$, $\text{GCF}(a,b) \times \text{LCM}(a,b)=ab$. If a statement gives the product and another gives the GCF, you can find the LCM, and vice versa.

However, knowing the GCF and LCM does not always determine the individual numbers. For example, the GCF 2 and LCM 12 are consistent with 4 and 6, and also with 2 and 12. Check whether the question asks for the pair or for a combination.

Some arithmetic statements are not sufficient alone but become sufficient when combined. Divisibility conditions often work this way. If one statement provides a factor of 2 and another provides a factor of 9, together they provide a factor of 18, provided the factors are independent. If both statements provide the same factor, combining them adds nothing.

Remainder conditions can also combine. If one statement gives a remainder modulo 3 and another gives a remainder modulo 4, together they can determine a remainder modulo 12. The Chinese remainder idea is not tested formally, but the pattern appears in questions about numbers that satisfy two conditions.

Data Sufficiency does not require the actual value. Once you know a statement determines one value or one consistent answer, you can stop. If a statement gives a unique remainder, you do not need to compute it. If a statement forces "yes" for every allowed case, you do not need to find the case.

This is a time-management skill as much as a reasoning skill. The test rewards the student who recognizes sufficiency quickly and moves on, not the one who solves every equation completely.

1. Decide whether the question asks for a value or a yes/no answer.
2. Evaluate each statement alone with several allowed values.
3. Test negatives, zero, and fractions when allowed.
4. Use algebra or prime factorization to confirm a pattern.
5. If neither statement works alone, combine them and test again.
6. Stop as soon as you can determine sufficiency. You do not need the actual value.

• Testing only positive integers when the question allows negatives.
• Assuming divisibility by 4 and 6 implies divisibility by 24.
• Forgetting that 2 is prime.
• Treating a remainder as a quotient.
• Using information from one statement while testing the other.
• Assuming two statements that give the same fact are independent.
• Computing a value when a yes/no answer is enough.