Practice Questions
Concept Explanation
When a is divided by b, we write a = bq + r where 0 ≤ r < b. Remainder questions appear frequently in placement 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 EasyWhat is the remainder when 47 is divided by 6?Explanation47 = 6 × 7 + 5. The remainder when 47 is divided by 6 is 5.
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Q2 EasyWhen 156 is divided by 13, what is the remainder?Explanation156 ÷ 13 = 12 exactly, so the remainder is 0.
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Q3 EasyWhat is the remainder when 10^100 is divided by 3?Explanation10 ≡ 1 (mod 3), so 10^100 ≡ 1^100 = 1. The remainder is 1.
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Q4 MediumFind the remainder when 2^10 is divided by 5.ExplanationPowers of 2 modulo 5 follow the cycle 2, 4, 3, 1 (length 4). Since 10 mod 4 = 2, the remainder is the second term: 4.
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Q5 MediumOn dividing a number by 5, we get 3 as remainder. What will the remainder when the square of the this number is divided by 5 ?ExplanationLet the number be x = 5k + 3. Then x² = (5k + 3)² = 25k² + 30k + 9 = 5(5k² + 6k + 1) + 4. On dividing x² by 5, the remainder is 4.
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Q6 MediumOn dividing a number by 357, we get 39 as remainder. On dividing the same number 17, what will be the remainder ?ExplanationLet the number be x = 357q + 39 = 17(21q + 2) + 5. Therefore, when x is divided by 17, the remainder is 5.
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Q7 MediumN is a whole number which when divided by 4 gives 3 as remainder. What will be the remainder when 2n is divided by 4 ?ExplanationLet n = 4q + 3. Then 2n = 8q + 6 = 4(2q + 1) + 2. Thus, when 2n is divided by 4, the remainder is 2.
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Q8 MediumWhat is the remainder when 123456 is divided by 9?ExplanationSum of digits = 1 + 2 + 3 + 4 + 5 + 6 = 21, and 2 + 1 = 3. The remainder on division by 9 is 3.
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Q9 MediumWhat is the remainder when 999 is divided by 7?Explanation999 = 7 × 142 + 5. Since 7 × 142 = 994, the remainder is 5.
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Q10 MediumA number when divided by 5 gives remainder 3. What remainder does its square leave when divided by 5?ExplanationIf n ≡ 3 (mod 5), then n² ≡ 9 ≡ 4 (mod 5). The square leaves remainder 4.
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Q11 HardWhat will be remainder when 17^200 is divided by 18 ?ExplanationWhen n is even, (x^n − a^n) is divisible by (x + a). So (17^200 − 1^200) is divisible by 18. On dividing 17^200 by 18, the remainder is 1.
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Q12 HardIn a division sum, the divisor is 10 times the quotient and 5 times the remainder. If the remainder is 46, what is the dividend ?ExplanationDivisor = 5 × 46 = 230. Quotient = 230 ÷ 10 = 23. Dividend = (Divisor × Quotient) + Remainder = (230 × 23) + 46 = 5336.
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Q13 HardWhen 2^50 is divided by 7, what is the remainder?Explanation2³ ≡ 1 (mod 7), so 2^50 = 2^(3×16+2) ≡ 2² = 4 (mod 7). The remainder is 4.
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Q14 HardThe remainder when 7^83 is divided by 5 is:Explanation7 mod 5 = 2. Powers of 2 mod 5 cycle as 2, 4, 3, 1. Since 83 mod 4 = 3, the remainder is 3.
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DS1 Easy
A positive integer N is given. What is the remainder when N is divided by 7?
Statement IN equals 45.
Statement IIWhen N is divided by 7, the remainder is 3.
ExplanationStatement I alone: 45 ÷ 7 = 6 remainder 3. Statement II alone describes the remainder but does not fix the value of N. -
DS2 Medium
What is the remainder when 2 raised to the power k is divided by 5?
Statement IThe exponent k equals 10.
Statement IIThe exponent k is an even positive integer.
Explanation2^10 mod 5: cycle length 4, 10 mod 4 = 2, so remainder is 4. Statement II alone (k even) does not fix the remainder.
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DS3 Medium
Is the positive integer N divisible by 3?
Statement IN leaves remainder 1 when divided by 3.
Statement IIN is a two-digit positive integer.
ExplanationStatement I alone shows N is not divisible by 3 (remainder 1). That directly answers the question. Statement II only restricts the size of N. -
DS4 Hard
What is the remainder when the positive integer N is divided by 4?
Statement IN equals 2 raised to the power 50.
Statement IIN is an even number.
Explanation2^50 mod 4: for exponent ≥ 2, 2^n ≡ 0 (mod 4), so the remainder is 0. Statement II alone (N is even) is insufficient.