Remainder: Formulas, Tricks and Solved Questions
Remainder is the value left after one number is divided by another. This page covers the division algorithm, remainder formulas, modular arithmetic, cyclicity of powers, divisibility-based shortcuts and the polynomial remainder theorem. Worked examples show how to solve common remainder questions involving large numbers, powers and algebraic expressions.
What is a Remainder?
The division algorithm is written as N = dq + r, where N is the dividend, d is the divisor, q is the quotient and r is the remainder. The conditions are 0 ≤ r < d. For example, 47 divided by 6 gives 47 = 6 × 7 + 5, so the remainder is 5.
Remainder Formula & Tricks
Important Formulas
Here, N is the dividend, d is the positive divisor, q is the quotient and r is the remainder.
Reduce each term by the divisor first, then add or subtract and take the non-negative remainder.
Each factor can be replaced by its remainder before multiplication.
Substitute x = a directly into the polynomial. For a divisor (x + a), substitute x = −a.
For n = qk + r, a^n ≡ a^r (mod m), provided the cycle is valid.
Quick Tricks
Replace a number by its remainder with respect to the divisor before raising it to a power.
For divisors such as 2^k, 5^k or 10^k, only the required last digits affect the remainder.
Find the smallest repeating pattern of powers modulo the divisor, then reduce the exponent by the cycle length.
Express the number in blocks or use the relation between the divisor and a convenient power of 10.
Remainder Concepts
Division Algorithm and Range of the Remainder
The remainder cannot be equal to or greater than the divisor. If division gives a negative intermediate value, add the divisor until the standard non-negative remainder is obtained. For example, 23 = 5 × 4 + 3, so 23 leaves remainder 3 when divided by 5.
Modular Arithmetic for Sums, Differences and Products
Congruences can be added, subtracted and multiplied. Thus, large expressions can be simplified by replacing each term with its remainder modulo m. The result is then reduced to the range 0 to m − 1.
Remainders of Large Powers
If a^k ≡ 1 (mod m), divide the exponent by k and use the remainder of that division. If the exponent is a multiple of k, the result is 1 modulo m. A shorter cycle may exist, so use the smallest valid cycle when possible.
Polynomial Remainder Theorem
This follows from P(x) = (x − a)Q(x) + P(a). For a linear divisor x + a, substitute −a. For a divisor of higher degree, ordinary polynomial division or factor-based methods are generally required.
Divisibility Rules as Remainder Tests
Useful tests include: divisibility by 3 or 9 depends on the digit sum; by 4 on the last two digits; by 8 on the last three digits; by 11 on the difference between sums of alternate digits; and by 2, 5 or 10 on the last digit.
Remainder Video Lessons
Watch short topic-wise lessons for quick revision.
Remainder Tricks: Fermat and Euler
Learn how Fermat’s pattern and Euler’s approach simplify remainder calculations for powers, including the key conditions and steps needed to apply these methods accurately.
Practice Remainder Questions
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Remainder Quick Quiz
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Remainder Revision Points
Use these rules to revise the main remainder methods and formulas.
- Write division as N = dq + r with 0 ≤ r < d.
- Reduce numbers modulo the divisor before adding, subtracting or multiplying.
- For powers, identify a repeating cycle and reduce the exponent using the cycle length.
- The remainder on division of P(x) by x − a is P(a).
- For divisor x + a, use P(−a) in the remainder theorem.
- A number is divisible by d when its remainder modulo d is zero.
- For divisors 10, 100 and 1000, use the last 1, 2 and 3 digits respectively.
- The standard remainder is non-negative and smaller than the positive divisor.
Remainder FAQs
What is the formula for finding a remainder?
Use N = dq + r, where N is the dividend, d is the divisor, q is the quotient and 0 ≤ r < d. Therefore, r = N − dq.
What is the remainder when 125 is divided by 7?
Since 125 = 7 × 17 + 6, the remainder is 6.
How is the remainder of a large power calculated?
Reduce the base modulo the divisor, find the repeating cycle of powers, and reduce the exponent using the cycle length. For example, 2^10 mod 7 = 2 because powers of 2 modulo 7 repeat every 3 terms and 10 mod 3 = 1.
What is the remainder theorem for a polynomial?
If P(x) is divided by x − a, the remainder is P(a). For example, the remainder of x^2 + 3x + 1 divided by x − 2 is P(2) = 4 + 6 + 1 = 11.
What is the remainder when a number is divided by 9?
The remainder equals the remainder obtained by dividing the sum of its digits by 9. For 4,738, the digit sum is 22, and 22 divided by 9 leaves remainder 4.
Can remainders be added or multiplied separately?
Yes. For modulus m, (a + b) mod m equals the sum of the individual remainders reduced modulo m, and (ab) mod m equals the product of the individual remainders reduced modulo m.