Practice Questions
Concept Explanation
When a is divided by b, we write a = bq + r where 0 ≤ r < b. Remainder questions appear frequently in TCS and Infosys aptitude tests.
Modular Arithmetic Basics
- (a + b) mod m = ((a mod m) + (b mod m)) mod m
- (a × b) mod m = ((a mod m) × (b mod m)) mod m
- If a ≡ b (mod m), then m divides (a − b).
Common Patterns
Remainders of powers cycle. For division by 9, sum of digits gives the remainder. For 5, use the last digit.
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Q1 Easy
What is the remainder when 47 is divided by 6?
Explanation47 = 6×7 + 5. Remainder is 5. -
Q2 Medium
Find the remainder when 2^10 is divided by 5.
ExplanationPowers of 2 mod 5 cycle: 2,4,3,1. 2^10: cycle length 4, 10 mod 4 = 2 → remainder 4. -
Q3 Easy
When 156 is divided by 13, the remainder is:
Explanation156 ÷ 13 = 12 exactly. Remainder 0. -
Q4 Hard
The remainder when 7^83 is divided by 5 is:
Explanation7 mod 5 = 2. Powers of 2 mod 5 cycle every 4: 2,4,3,1. 83 mod 4 = 3 → remainder 3. -
Q5 Medium
If a number leaves remainder 4 when divided by 7, which could be the number?
Explanation32 ÷ 7 = 4 remainder 4. The others leave remainders 4, 2, 0 respectively. -
Q6 Medium
What is the remainder when 123456 is divided by 9?
ExplanationDigit sum = 21, 2+1=3. Remainder when divided by 9 is 3. -
Q7 Hard
The smallest positive number that leaves remainder 2 when divided by 3 and 4 is:
ExplanationNumbers ≡ 2 (mod 3) and (mod 4): check 2(no), 5(no), 8(no mod3), 14: 14 mod 3=2, 14 mod 4=2. Answer 14.
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Q8 Medium
Remainder when (15 × 23) is divided by 8:
Explanation15 mod 8 = 7, 23 mod 8 = 7. Product 49 mod 8 = 1. -
Q9 Easy
If N leaves remainder 1 when divided by 4, N could be:
Explanation25 ÷ 4 = 6 remainder 1. The other options leave remainders 2, 2, and 2. -
Q10 Medium
What is the remainder when 999 is divided by 7?
Explanation999 = 7×142 + 5? 7×142=994, remainder 5. Answer C=5. -
Q11 Hard
When 2^50 is divided by 7, the remainder is:
Explanation2^3 ≡ 1 (mod 7), so 2^50 = 2^(3×16+2) ≡ 2² = 4 (mod 7). -
Q12 Medium
Sum of remainders when 25 and 37 are each divided by 8:
Explanation25 mod 8 = 1, 37 mod 8 = 5. Sum = 6. -
Q13 Medium
A number when divided by 5 gives remainder 3. Its square gives remainder:
ExplanationIf n ≡ 3 (mod 5), n² ≡ 9 ≡ 4 (mod 5). -
Q14 Easy
The remainder when 10^100 is divided by 3 is:
Explanation10 ≡ 1 (mod 3), so 10^100 ≡ 1^100 = 1.
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DS1 Easy
What is the remainder when N is divided by 7?
Statement IN = 45.
Statement IIN leaves remainder 3 when divided by 7.
Explanation45 ÷ 7 = 6 remainder 3. Statement I alone suffices. Statement II alone does not fix N. -
DS2 Medium
What is the remainder when 2^k is divided by 5?
Statement Ik = 10.
Statement IIk is even.
Explanation2^10 mod 5: cycle length 4, 10 mod 4 = 2 → remainder 4. Statement II alone (k even) does not fix the remainder.
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DS3 Medium
Is N divisible by 3?
Statement IN leaves remainder 1 when divided by 3.
Statement IIN is a two-digit number.
ExplanationStatement I alone tells us N is not divisible by 3 (remainder 1). That answers the question. Statement II only restricts size. -
DS4 Hard
What is the remainder when N is divided by 4?
Statement IN = 2^50.
Statement IIN is even.
Explanation2^50 mod 4: for exponent ≥ 2, 2^n ≡ 0 (mod 4). Remainder 0. Statement II alone is insufficient.