Different Types of Remainder Problems
Different Types of Remainder Problems include direct division, powers, algebraic expressions, common remainders and divisibility-based questions. Use the division algorithm, modular arithmetic and remainder cycles to simplify large calculations. This page presents the main formulas, methods, examples and shortcuts used to solve remainder problems in quantitative aptitude.
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What Are Different Types of Remainder Problems?
Here, q is the quotient and r is the remainder. In modular notation, N ≡ r (mod d). For example, 47 = 5 × 9 + 2, so the remainder when 47 is divided by 5 is 2. All calculations involving remainders must be reduced to a value between 0 and d − 1.
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Different Types of Remainder Problems
Learn how to solve different types of remainder problems in the Number System, including the key methods and steps needed to determine remainders accurately.
Different Types of Remainder Problems Formula & Tricks
Important Formulas
N is the dividend, d is the divisor, q is the quotient and r is the remainder.
Replace each number by its remainder before adding or subtracting, then reduce the result to the standard remainder.
The factors can be reduced separately before multiplication.
For large powers, first reduce the base and then use the repeating remainder cycle.
Every number that leaves remainder r on division by d has this form, where k is an integer.
Quick Tricks
In addition, subtraction and multiplication, replace each term by its remainder with respect to the divisor. This keeps the calculation small.
For division by 2, 4, 5, 8, 10 or powers related to 10, only the required last digits may be needed.
The remainders of powers often repeat. Find the cycle length, divide the exponent by that length and use the corresponding cycle position.
If a calculation gives a negative value, add the divisor until the result lies between 0 and d − 1.
Different Types of Remainder Problems Concepts
Direct Remainder Using the Division Algorithm
If N = dq + r, the remainder is r. The remainder is always smaller than the divisor. For example, 125 = 12 × 10 + 5, so the remainder when 125 is divided by 10 is 5.
Remainders of Sums, Differences and Products
For divisor d, reduce each number modulo d before calculation. If a mod d = r₁ and b mod d = r₂, then a + b has remainder (r₁ + r₂) mod d, while ab has remainder (r₁r₂) mod d.
Remainders of Powers and Cyclicity
For example, powers of 3 modulo 7 are 3, 2, 6, 4, 5, 1, repeating every 6 powers. If the exponent is a multiple of 6, the remainder is 1. Otherwise, use the exponent's remainder after division by 6.
Common Remainder Problems
For divisors d₁, d₂ and d₃, N − r must be divisible by each divisor. Hence, N − r is divisible by lcm(d₁, d₂, d₃). The smallest positive number is obtained by taking the least suitable value of k.
Remainder of Algebraic Expressions
If x leaves remainder r when divided by d, then x ≡ r (mod d). Therefore, x² + 3x + 5 can be evaluated as r² + 3r + 5 modulo d. The final value must be reduced to the standard remainder.
Different Types of Remainder Problems Data Table
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Common Remainder Problem Types
Rules used for frequently occurring remainder calculations
| Problem type | Method | Example |
|---|---|---|
| Direct division | Use N = dq + r | 47 divided by 5 gives remainder 2 |
| Sum or difference | Reduce each term modulo d, then operate | (38 + 47) mod 7 = (3 + 5) mod 7 = 1 |
| Product | Multiply the individual remainders | (38 × 47) mod 7 = (3 × 5) mod 7 = 1 |
| Large power | Find the repeating cycle | 2^100 mod 7 = 2 |
| Same remainder | Use lcm of the divisors and add the common remainder | lcm(3, 4, 5) + 2 = 62 |
Note: In every case, the final remainder must be between 0 and divisor − 1.
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Remainder Problems Revision Points
Use these rules to revise the main remainder methods.
- Division algorithm: N = dq + r, where 0 ≤ r < d.
- Remainders can be reduced before addition, subtraction and multiplication.
- For powers, find the repeating cycle of remainders.
- If a number leaves the same remainder r for several divisors, subtract r and use the LCM of the divisors.
- If x ≡ r (mod d), any polynomial in x can be evaluated using r instead of x.
- A negative modular result is converted to the standard remainder by adding the divisor.
- The remainder is never equal to or greater than the divisor.
Different Types of Remainder Problems FAQs
What is the basic formula for a remainder?
The formula is N = dq + r, where N is the dividend, d is the divisor, q is the quotient and 0 ≤ r < d.
What is the remainder when 2^100 is divided by 7?
The powers of 2 modulo 7 repeat as 2, 4, 1. Since 100 mod 3 = 1, the remainder is 2.
What is the remainder when 234 × 157 is divided by 11?
234 mod 11 = 3 and 157 mod 11 = 3. Therefore, 234 × 157 mod 11 = 3 × 3 mod 11 = 9.
How do you find the smallest number that leaves the same remainder on division by several numbers?
Subtract the common remainder from the number. The result must be divisible by the LCM of all the divisors. Thus, the smallest number is usually LCM of the divisors plus the common remainder.
What is the smallest number that leaves remainder 3 when divided by 4, 6 and 8?
The LCM of 4, 6 and 8 is 24. Therefore, the smallest number is 24 + 3 = 27.
How is a negative remainder converted into a standard remainder?
Add the divisor until the result lies between 0 and the divisor minus 1. For example, −3 modulo 7 equals 4.