Integer Calculator
Free Integer Calculator for arithmetic. Enter values to get step-by-step solutions with formulas and graphs. Enter your values for instant results.
Formula
a op b = result | a = bq + r (division algorithm)
Integer arithmetic performs operations on whole numbers. For division, the division algorithm states that for any integers a and b (b not zero), there exist unique integers q (quotient) and r (remainder) such that a = bq + r, where 0 <= r < |b|.
Worked Examples
Example 1: Integer Division with Remainder
Problem: Divide 157 by 12 using integer division.
Solution: 157 / 12 = 13 remainder 1\nVerification: 12 x 13 + 1 = 156 + 1 = 157\nQuotient = 13\nRemainder = 1\n157 mod 12 = 1
Result: 157 = 12 x 13 + 1 (Quotient: 13, Remainder: 1)
Example 2: Integer Arithmetic Operations
Problem: Calculate all basic operations for -24 and 7.
Solution: Sum: -24 + 7 = -17\nDifference: -24 - 7 = -31\nProduct: -24 x 7 = -168\nQuotient: -24 / 7 = -4 remainder 4\nGCD: GCD(24, 7) = 1 (coprime)\nLCM: LCM(24, 7) = 168
Result: Sum = -17 | Product = -168 | GCD = 1 | LCM = 168
Frequently Asked Questions
What is integer division and how does the remainder work?
Integer division divides one integer by another and returns only the whole number part of the quotient, discarding any fractional remainder. For example, 17 divided by 5 gives quotient 3 and remainder 2, because 5 times 3 equals 15 and 17 minus 15 equals 2. This is expressed as 17 = 5 times 3 + 2. The division algorithm states that for any integers a and b (with b not zero), there exist unique integers q and r such that a equals b times q plus r, where 0 is less than or equal to r which is less than the absolute value of b. This concept is foundational for modular arithmetic and the Euclidean algorithm.
What does it mean for an integer to be even or odd?
An even integer is divisible by 2 with no remainder, meaning it can be written as 2k for some integer k. Examples include -4, 0, 2, 8, 100. An odd integer leaves a remainder of 1 when divided by 2 and can be written as 2k+1. Examples include -3, 1, 7, 99. Important properties: even plus even equals even, odd plus odd equals even, even plus odd equals odd. For multiplication: even times any integer equals even, odd times odd equals odd. Zero is considered even because 0 equals 2 times 0. These parity properties are used extensively in proofs and computer science.
What is the absolute value of an integer?
The absolute value of an integer is its distance from zero on the number line, always expressed as a non-negative number. It is denoted with vertical bars: |n|. For positive numbers and zero, the absolute value equals the number itself: |5| equals 5 and |0| equals 0. For negative numbers, the absolute value is the positive counterpart: |-7| equals 7. Absolute value satisfies several important properties: |ab| equals |a| times |b|, |a+b| is less than or equal to |a| plus |b| (triangle inequality), and |a| equals 0 if and only if a equals 0. Absolute value is essential in defining distance and error metrics.
What are some important properties of integer arithmetic?
Integer arithmetic has several fundamental properties. Closure: the sum, difference, and product of any two integers is always an integer (but not necessarily the quotient). Commutativity: a plus b equals b plus a, and a times b equals b times a. Associativity: (a plus b) plus c equals a plus (b plus c). Distributivity: a times (b plus c) equals a times b plus a times c. Identity elements: 0 for addition and 1 for multiplication. Additive inverse: for every integer a, there exists negative a such that a plus negative a equals 0. These properties make the integers a commutative ring in abstract algebra.
How accurate are the results from Integer Calculator?
All calculations use established mathematical formulas and are performed with high-precision arithmetic. Results are accurate to the precision shown. For critical decisions in finance, medicine, or engineering, always verify results with a qualified professional.
How do I interpret the result?
Results are displayed with a label and unit to help you understand the output. Many calculators include a short explanation or classification below the result (for example, a BMI category or risk level). Refer to the worked examples section on this page for real-world context.