Remainder questions look like number theory, but most of them reward one practical idea: you can work with remainders instead of the full numbers. On the GMAT Focus they show up in Quant Problem Solving and, very often, in Data Insights Data Sufficiency, where the question is whether a remainder is fixed at all.
Write the relationship explicitly. “n leaves remainder 3 when divided by 7” means n = 7k + 3 for some integer k ≥ 0, so n can be 3, 10, 17, 24 and so on. Listing the first few values is often the fastest route, especially when two conditions must hold at once: list values for each and look for the first overlap. The overlaps then repeat every LCM of the two divisors.
Remainders add and multiply. The remainder of a sum or a product depends only on the remainders of the pieces, so large expressions shrink quickly. For powers, find the cycle of last digits or remainders and use the position in the cycle.
Watch the edges. A remainder is always smaller than the divisor, it can be zero, and a smaller number divided by a larger one leaves itself as the remainder: 5 divided by 8 gives quotient 0, remainder 5. In Data Sufficiency, test the smallest values that fit before deciding a statement is sufficient.
4 practice questions
Problem SolvingRemaindersMedium
Question 1
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.
Data SufficiencyRemaindersMedium
Question 2
If $n$ is a positive integer, what is the remainder when $n$ is divided by 6?
(1) When $n$ is divided by 12, the remainder is 7.
(2) $n + 5$ is divisible by 6.
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: D
Key Concept or Approach
If the divisor in a statement is a multiple of 6, the remainder mod 6 follows by reducing that statement's remainder mod 6.
Step-by-Step Solution
Step 1: Statement (1): $n = 12k + 7 = 6(2k + 1) + 1$, so the remainder is 1. Sufficient.
Step 2: Statement (2): $n + 5 = 6m$, so $n = 6m - 5 = 6(m - 1) + 1$, and the remainder is 1. Sufficient.
Step 3: Each statement alone is sufficient.
Final Answer
Answer: D
Common trap: B: students think a remainder of 7 on division by 12 says nothing about division by 6, even though 12 is a multiple of 6.
Data SufficiencyRemaindersMedium
Question 3
If $n$ is a positive integer, what is the remainder when $n$ is divided by $5$?
(1) When $3n$ is divided by $5$, the remainder is $4$.
(2) When $n^2$ is divided by $5$, the remainder is $4$.
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: A
Key Concept or Approach
Test each possible remainder $r \in \{0, 1, 2, 3, 4\}$ of $n$: the remainders of $3n$ and $n^2$ depend only on $r$.
Step 2: Statement (1): only $r = 3$ makes $3n$ leave remainder $4$ (e.g. $n = 3$: $9 = 5 + 4$; $n = 8$: $24 = 20 + 4$). Sufficient: the remainder is $3$.
Step 3: Statement (2): $r = 2$ and $r = 3$ both work ($n = 2$: $n^2 = 4$; $n = 3$: $n^2 = 9 = 5 + 4$). Remainder could be $2$ or $3$. Not sufficient.
Final Answer
Answer: A
Common trap: D: students assume both statements 'reverse' to a single remainder; squaring loses information because 2 and 3 have the same square mod 5.
Data SufficiencyRemaindersHard
Question 4
If $n$ is a positive integer, what is the remainder when $n$ is divided by $24$?
(1) When $n$ is divided by $8$, the remainder is $5$.
(2) When $n$ is divided by $6$, the remainder is $1$.
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
A remainder mod $24$ is determined when the conditions pin down $n$ mod $\text{lcm}$ $= 24$. List candidates below $24$.
Step-by-Step Solution
Step 1: Statement (1): $n = 8k + 5$ leaves remainder $5$, $13$ or $21$ on division by $24$ ($n = 5$ gives $5$, $n = 13$ gives $13$). Not sufficient.
Step 3: Together: the only common value among $0$-$23$ is $13$ ($13 = 8 + 5 = 12 + 1$). Since $\text{lcm}(8, 6) = 24$, every such $n$ is $13$ more than a multiple of $24$. Remainder $13$. Sufficient.
Final Answer
Answer: C
Common trap: E: students multiply 8 x 6 = 48, decide the combined pattern repeats every 48 not 24, and expect two remainders.
