DATA INSIGHTS · GMAT-DI-DSNP · DRAFT

Data Sufficiency: Number Properties

Build the concept, interpret evidence, and apply it in a separate assessment pool.

Learn: Evidence and decisions

Test what each factor condition guarantees

A multiple of 2 is not automatically a multiple of 6, and a multiple of 3 is not automatically a multiple of 6. Check each statement against the full target.

Combine coprime requirements

For a positive integer, being divisible by 2 and by 3 guarantees divisibility by 6 because the factors are coprime.

Translate divisibility into a structure

For a remainder question, write n=dq+r with 0≤r<d. For parity and factor questions, test the smallest admissible values and check whether the condition rules out counterexamples.

Respect integer and positivity conditions

A statement may be sufficient only because the prompt restricts the variable to integers or positive integers. Use every stated domain condition.

Learning Objectives

Use prime factors and divisibility to assess sufficiency.
Separate divisibility by factors from divisibility by their product.
Find allowed counterexamples for each statement.

Prerequisite Knowledge

Prime factorization.
For coprime integers a and b, divisibility by both implies divisibility by ab.

Worked Examples

Shared factors do not multiply automatically

A positive whole number of samples can be packed into groups of four and into groups of six without leftovers. Is the count necessarily divisible by twenty-four? Evaluate the group-of-four condition and the group-of-six condition as separate statements.

The first statement allows both of the following counts, so it is insufficient. $n = 4 or 24$
The second also permits a No case and a Yes case. $n = 6 or 24$
Together, the count must be a multiple of the least common multiple, not necessarily the product. $lcm (4, 6) = 12$
Both counts satisfy both statements, but only one answers Yes. $\frac{12}{24} = \frac{1}{2}, \frac{24}{24} = 1$

Conclusion: Even together the statements are insufficient: twelve samples and twenty-four samples give different answers. Shared prime factors must not be counted twice.

Remainder form isolates the answer

What is the remainder when positive integer n is divided by 7? (1) n=7k+3 for some integer k. (2) n is odd.

Statement (1) is already in quotient-remainder form. Since 3 is between 0 and 6, the remainder is fixed. $n = 7 k + 3 R i g h t a r r o w r = 3$
Oddness alone leaves several possible remainders, including 1, 3 and 5.

Conclusion: Statement (1) alone is sufficient and statement (2) alone is not. The answer is (A).

Use the domain to prove a factor claim

Is positive integer n composite? (1) n is odd. (2) n is a positive multiple of 9.

Oddness alone does not decide: 3 is prime, while 9 is composite.
Every positive multiple of 9 is at least 9 and has nontrivial factors 3 and n/3. $9 mi d n R i g h t a r r o w n = 9 k = 3 (3 k), q u a d k g e q 1$

Conclusion: Statement (2) alone guarantees that n is composite, whereas statement (1) does not. The answer is (B).

Guided Practice: Try, then check

Work through each question on paper before opening its hint or explanation. This learning activity is separate from the scored Practice and Simulation pools.

Guided case: preserve the required prime powers

A positive whole number of samples must be packed without leftovers into groups of twenty. Can this be done?

Sortable data table. Select a column heading to sort.

(1)	The count is divisible by ten.
(2)	The count is divisible by four.

Find opposite answers allowed by statement one.
Show hint
Use a small multiple of ten and then the target group size.
Check reasoning
Ten samples cannot form a full group of twenty, whereas twenty can. Both satisfy the first statement.
$10 \neq \equiv 0 (mod 20), 20 \equiv 0 (mod 20)$
Find opposite answers allowed by statement two, without borrowing statement one.
Show hint
A multiple of four need not be a multiple of ten.
Check reasoning
Four samples and twenty samples both satisfy statement two, but only twenty can be packed as required.
Combine the prime-factor requirements. What is guaranteed?
Show hint
Use the largest required power of each prime, not the product of the divisors.
Check reasoning
The combined count must contain two factors of two and one factor of five, so it is a multiple of twenty. Both statements together are sufficient.
$10 = 2 \cdot 5, 4 = 2^{2}, lcm (10, 4) = 2^{2} \cdot 5 = 20$

When combining divisibility conditions, preserve prime powers. Multiplying divisors can overstate what is guaranteed.

Think Like a GMAT Test-Taker

Factor the target before evaluating the statements.

Use a small multiple of the stated factor as a counterexample.

When combining conditions, confirm whether the factors are coprime.

Common Traps

Mistaking divisibility by one factor for divisibility by the product.

Assuming n is prime or has no other factors.

Checking only large examples.