Number properties questions test how integers behave: even and odd, primes, divisibility, factors, multiples and remainders. They are among the most common Quant and Data Sufficiency topics, and they reward understanding over computation. Many can be answered without doing any real arithmetic.
Build your toolkit around prime factorization. Once a number is written as a product of primes, you can count its divisors (add one to each exponent and multiply), test divisibility, and find greatest common factors and least common multiples. For remainders, you can replace a number by its remainder: if n leaves remainder 4 when divided by 7, then 3n + 5 leaves the same remainder as 3 × 4 + 5.
The traps are hidden cases. Two is the only even prime, and it breaks every “p is prime, so p is odd” shortcut. The product of two consecutive integers is always even, so a statement about n(n + 1) says nothing about n itself. Zero is even, and negative integers have parity too. In Data Sufficiency, test the smallest cases on purpose (1, 2, a prime, a perfect square) before deciding a statement is sufficient.
The four questions below mix Problem Solving and Data Sufficiency, from a quick parity check to harder divisor counting.
4 practice questions
Data SufficiencyNumber PropertiesEasy
Question 1
If $n$ is an integer, is $n$ even?
(1) $n^2 + n$ is even.
(2) $n + 3$ is odd.
AStatement (1) ALONE is sufficient, but statement (2) ALONE is NOT sufficient.
BStatement (2) ALONE is sufficient, but statement (1) ALONE is NOT sufficient.
CBOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
DEACH statement ALONE is sufficient.
EStatements (1) and (2) TOGETHER are NOT sufficient.
Show answer & explanation
Correct answer: B
Key Concept or Approach
Parity rules: odd + odd = even, even + odd = odd. $n^2 + n = n(n+1)$ is a product of consecutive integers, so it is always even.
Step-by-Step Solution
Step 1: Statement (1): $n^2 + n = n(n+1)$ is even for every integer $n$. If $n = 2$ the answer is yes; if $n = 3$ ($9 + 3 = 12$, even) the answer is no. Not sufficient.
Step 2: Statement (2): $n + 3$ is odd and 3 is odd, so $n$ = odd − odd = even. Sufficient.
Step 3: Only statement (2) is sufficient.
Final Answer
Answer: B
Common trap: D: students think "$n^2 + n$ even" forces $n$ even, forgetting that $n(n+1)$ is always even.
Problem SolvingRemaindersMedium
Question 2
When the positive integer $n$ is divided by $7$, the remainder is $4$. What is the remainder when $3n + 5$ is divided by $7$?
A$0$
B$1$
C$2$
D$3$
E$4$
Show answer & explanation
Correct answer: D
Key Concept or Approach
Write $n = 7q + 4$ (or test $n = 4$); remainders pass through multiplication and addition.
Step-by-Step Solution
Step 1: $3n + 5 = 3(7q+4)+5 = 21q + 17$.
Step 2: $21q$ is a multiple of 7, and $17 = 14 + 3$.
Step 3: The remainder is $3$ (check: $n=11$ gives $38 = 35 + 3$).
Final Answer
Answer: D
Common trap: 2 - adding 4 + 5 = 9 and forgetting to multiply the remainder by 3.
Problem SolvingDivisibility/Multiples/FactorsHard
Question 3
How many positive divisors of $7{,}200$ are multiples of $12$?
A$18$
B$24$
C$27$
D$36$
E$54$
Show answer & explanation
Correct answer: B
Key Concept or Approach
Prime-factorize, then force the divisor to contain $12 = 2^2 \cdot 3$.
Step-by-Step Solution
Step 1: $7200 = 2^5 \cdot 3^2 \cdot 5^2$.
Step 2: A divisor $2^a3^b5^c$ is a multiple of 12 iff $a \ge 2$ and $b \ge 1$: $a \in \{2,3,4,5\}$, $b \in \{1,2\}$, $c \in \{0,1,2\}$.
Step 3: Count: $4 \times 2 \times 3 = 24$.
Final Answer
Answer: B
Common trap: 54 - the total number of divisors of 7,200, ignoring the multiple-of-12 condition.
Data SufficiencyDivisibility/Multiples/FactorsNumber PropertiesVery hard
Question 4
If $n$ is a positive integer, how many positive divisors does $n$ have?
(1) $n = 2p$, where $p$ is a prime number.
(2) $n$ is divisible by 4.
AStatement (1) ALONE is sufficient, but statement (2) ALONE is NOT sufficient.
BStatement (2) ALONE is sufficient, but statement (1) ALONE is NOT sufficient.
CBOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
DEACH statement ALONE is sufficient.
EStatements (1) and (2) TOGETHER are NOT sufficient.
Show answer & explanation
Correct answer: C
Key Concept or Approach
The divisor count depends on the prime factorisation. $2p$ has 4 divisors when $p$ is an odd prime but only 3 when $p = 2$ (since $n = 4 = 2^2$).
