Integer Calculator
Perform arithmetic operations on integers including addition, subtraction, multiplication, division with remainder, and exponentiation.
Integer Calculator
Calculator
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Formula: a op b = result | a = bq + r (division algorithm)
Worked example — 157 = 12 x 13 + 1 (Quotient: 13, Remainder: 1)
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 Verification: 12 x 13 + 1 = 156 + 1 = 157 Quotient = 13 Remainder = 1 157 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 Difference: -24 - 7 = -31 Product: -24 x 7 = -168 Quotient: -24 / 7 = -4 remainder 4 GCD: GCD(24, 7) = 1 (coprime) LCM: LCM(24, 7) = 168
Result:Sum = -17 | Product = -168 | GCD = 1 | LCM = 168
Frequently Asked Questions
What are integers and how are they different from other numbers?
Integers are whole numbers that can be positive, negative, or zero. They include numbers like -3, -2, -1, 0, 1, 2, 3 and extend infinitely in both directions on the number line. Unlike fractions or decimals, integers have no fractional or decimal parts. They are different from natural numbers (which start at 1 or 0 depending on convention) and from rational numbers (which include fractions). Integers form a fundamental set in mathematics denoted by the symbol Z, from the German word Zahlen meaning numbers. They are closed under addition, subtraction, and multiplication.
How do you add and subtract integers with different signs?
When adding integers with the same sign, add their absolute values and keep the sign. For example, -5 plus -3 equals -8. When adding integers with different signs, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value. So -7 plus 4 equals -3 because 7 minus 4 is 3 and the negative sign dominates. For subtraction, convert it to addition by changing the sign of the second number: a minus b equals a plus negative b. So 5 minus (-3) becomes 5 plus 3 which equals 8. These rules follow from the number line model.
What are the rules for multiplying and dividing integers?
The sign rules for multiplication and division are the same: positive times positive equals positive, negative times negative equals positive, and positive times negative (or vice versa) equals negative. An easy mnemonic is that an even number of negative factors gives a positive product, while an odd number gives a negative product. For division, the same rules apply but remember that integer division may produce a non-integer result. In integer arithmetic, we often use floor division (rounding toward negative infinity) or truncation (rounding toward zero), along with the remainder. Division by zero is always undefined.
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 prime numbers and composite numbers among integers?
A prime number is a positive integer greater than 1 that has exactly two distinct positive divisors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Note that 2 is the only even prime. A composite number is a positive integer greater than 1 that has more than two positive divisors. For example, 12 is composite because it is divisible by 1, 2, 3, 4, 6, and 12. The number 1 is neither prime nor composite by convention. The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be uniquely expressed as a product of primes.
How do powers and exponents work with integers?
When raising an integer to a positive integer power, you multiply the base by itself that many times: 3 to the 4th equals 3 times 3 times 3 times 3 equals 81. Any non-zero integer to the power of 0 equals 1. Negative exponents give reciprocals: 2 to the -3 equals 1/8, which is no longer an integer. For negative bases, even exponents produce positive results (negative 3 squared equals 9) while odd exponents produce negative results (negative 3 cubed equals negative 27). Powers grow extremely fast, which is why exponential growth is so powerful in both mathematics and practical applications like compound interest.
What is modular arithmetic and how does it relate to integers?
Modular arithmetic is a system of arithmetic for integers where numbers wrap around after reaching a certain value called the modulus. We say a is congruent to b modulo m if m divides (a minus b). For example, 17 is congruent to 2 modulo 5 because 17 minus 2 equals 15, which is divisible by 5. Clock arithmetic is a familiar example: 14 hours after 11 o'clock is 1 o'clock (14 plus 11 equals 25, and 25 mod 12 equals 1). Modular arithmetic is crucial in cryptography (RSA), hash functions, check digits (ISBN, credit cards), and computer science algorithms.
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.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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