Integers: Number Line, Operations, Divisibility, GCD and LCM

What are integers?

Integers are the numbers with no fractional part: the counting numbers 1,2,3,…, their negatives -1,-2,-3,… and zero. The set is written ℤ, from the German Zahlen ("numbers").

Negative numbers exist because people need to describe things that go below a starting point, and the integers are what you get when a quantity can go either way:

Three kinds of number are often confused. The natural numbers are 1,2,3,… (here they start at 1; some books start at 0). The whole numbers are 0,1,2,3,…. The integers add the negatives. Every natural number is a whole number and every whole number is an integer, but not the other way round; the article on real numbers shows how all these sets nest.

Two facts about zero and signs are worth fixing now. Zero is neither positive nor negative. And every integer a has an opposite, -a, on the other side of 0 at the same distance, so that a+(-a)=0. The opposite of 7 is -7, the opposite of -7 is 7, and the opposite of 0 is 0, which is why -(-a)=a.

The integers are a closed family under three operations: add, subtract or multiply two integers and the result is always an integer. They are not closed under division, since 7÷2 is not an integer. That one gap is what the later sections on remainders are about, and what the article on rational numbers closes with fractions.

How do you compare integers and what is absolute value?

On the number line the integer further to the right is the greater one, so -3>-7 even though 7 is bigger than 3. Every positive integer is greater than 0 and every negative one is less than 0, which makes every positive integer greater than every negative one.

To compare two negative numbers, compare their sizes and reverse the answer: 7>3, so -7<-3. A debt of 7 is worse than a debt of 3.

The absolute value |a| is the distance from a to 0, so it is never negative: |7|=7, |-7|=7 and |0|=0. In symbols, |a|=a when a\ge0 and |a|=-a when a<0. Opposites have the same absolute value, and the distance between two integers is the absolute value of their difference: |-4-3|=7.

Ordering a list is a matter of reading it left to right on the line: negatives first, with the largest size furthest left, then 0, then the positives.

How do you add and subtract integers?

To add integers, move along the number line: adding a positive moves right and adding a negative moves left; to subtract, add the opposite, so a-b=a+(-b). That one rule covers every case.

Two rules for adding, by the signs:

• Same signs: add the sizes and keep the sign. 5+3=8 and (-5)+(-3)=-8.
• Different signs: subtract the smaller size from the larger, and keep the sign of the larger. 5+(-8)=-3, because 8 is the larger size and it is negative. (-5)+8=3.

Two numbers that are opposites cancel: 6+(-6)=0.

Subtracting is adding the opposite. Taking away a positive moves left: 2-5=2+(-5)=-3. Taking away a negative moves right, because removing a debt makes you richer: 5-(-3)=5+3=8 and (-5)-(-3)=-5+3=-2.

The order does not matter for adding, a+b=b+a, but it does for subtracting: 5-3=2 while 3-5=-2. Try your own numbers below; the first jump (above the line) goes from 0 to a and the second (below) is the change.

How do you multiply and divide integers?

Multiply or divide the sizes, then make the answer positive if the signs are the same and negative if they differ. A product with a factor of 0 is 0.

Division follows the same signs, because a÷ b=c means a=b· c: (-12)÷4=-3 since 4·(-3)=-12, and (-12)÷(-3)=4 since (-3)·4=-12.

Why is a negative times a negative positive? Two arguments show it is not a convention but forced.

A pattern. Count down the first factor and watch the products: 3·(-2)=-6, 2·(-2)=-4, 1·(-2)=-2, 0·(-2)=0. Each step down adds 2. The next step must add 2 again: (-1)·(-2)=2.

The distributive law. Start from 0=(-1)·0=(-1)\bigl(1+(-1)\bigr)=(-1)·1+(-1)(-1)=-1+(-1)(-1). For the right side to equal 0, (-1)(-1) has to be 1. Any other value would break the distributive law, a(b+c)=ab+ac, that the whole of arithmetic depends on.

Division by zero is undefined. a÷0=c would mean a=0· c=0, which is impossible for a≠0; and 0÷0 could be any number, so it has no single value. Meanwhile 0÷ a=0 for a≠0.

Integers are not closed under division: 7÷2 is the fraction \frac72, which is why division with remainder is its own topic below.

What is the order of operations with negative numbers?

Work in this order: brackets, then powers, then multiplication and division from left to right, then addition and subtraction from left to right. Negative numbers add two traps to it.

1. A minus sign is not a bracket. -3^2 means -(3^2)=-9, because the power binds tighter than the minus. Only (-3)^2=9 squares the -3.
2. Same-level operations go left to right. 12÷(-3)×2 is -4×2=-8, not 12÷(-6)=-2. And 10-4-3=3, not 10-(4-3)=9.
3. Put brackets round a negative after an operator. 5·-2 is written 5·(-2), and 8-3×(-2)=8+6=14: the multiplication goes before the subtraction.

An even power of a negative number is positive and an odd power is negative: (-2)^4=16 and (-2)^3=-8, since each pair of minus signs cancels.

What are even and odd numbers and the divisibility rules?

An integer a is divisible by b when a=b· k for some integer k: b is then a divisor (factor) of a and a is a multiple of b. So 12 is divisible by 4 because 12=4·3, and 0 is divisible by every integer because 0=b·0.

Even and odd. An even integer is divisible by 2, a=2k; an odd one is a=2k+1. Zero is even, and negative numbers have parity too: -3 is odd and -4 is even. The even numbers 2,4,6,…, the odd numbers 1,3,5,… and the multiples of any integer each form an arithmetic sequence, and their sums have a formula there. Parity follows a short table:

A proof of one row: (2a+1)+(2b+1)=2(a+b+1), which is even. The same style of proof shows that the square of an odd number is odd, the fact behind the irrationality of √2.

Divisibility rules let you test a number from its digits, without dividing:

Example: 7200 has digit sum 9 and ends in 00, so it is divisible by 2, 3, 4, 5, 6, 8, 9 and 10. For 11 take the digits from the right with alternating signs, 0-0+2-7=-5, which is not a multiple of 11, so 7200 is not. Try any number below, including a very long one.

What are prime numbers and prime factorization?

A prime number is an integer greater than 1 whose only positive divisors are 1 and itself; every integer greater than 1 is either prime or a product of primes, and that product is unique. The primes begin 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47, fifteen of them below 50.

• Composite numbers have more divisors: 12=3·4.
• 1 is neither prime nor composite. It has one divisor, itself. Calling it prime would break uniqueness, since 6=2·3=1·2·3.
• 2 is the only even prime, because every other even number is also divisible by 2.

The fundamental theorem of arithmetic says every integer greater than 1 can be written as a product of primes in exactly one way, apart from the order. So 360=2·2·2·3·3·5=2^3·3^2·5, and no other set of primes multiplies to 360.

To factorize, divide by the smallest prime that goes in, again and again, until 1 is left: 360÷2=180, ÷2=90, ÷2=45, ÷3=15, ÷3=5, ÷5=1.

To test whether n is prime, divide by the primes up to √n. If n had a factor larger than √n, its partner factor would be smaller than √n, so you would have met it already. Take 91: √(91)<10, and 91=7·13, so it only looks prime. For 97 nothing in 2, 3, 5, 7 divides it, so 97 is prime.

There are infinitely many primes. Euclid's argument: suppose a list of primes were complete, multiply them all and add 1. The result leaves remainder 1 on division by every prime on the list, so it either is a new prime or has one, and the list was not complete.

Divisors from the factorization. If n=p^a q^b… it has (a+1)(b+1)… positive divisors, so 360 has 4·3·2=24. Use the tool below.

How do you find the GCD and LCM?

The greatest common divisor (GCD, in Indonesian FPB) of two integers is the largest integer that divides both, and the least common multiple (LCM, KPK) is the smallest positive integer that both divide. For 48 and 180 the GCD is 12 and the LCM is 720.

By prime factors. Write each number as primes: 48=2^4·3 and 180=2^2·3^2·5. For the GCD take each prime with the smaller exponent: 2^2·3=12. For the LCM take each prime with the larger exponent: 2^4·3^2·5=720.

By Euclid's algorithm, which needs no factoring and works for huge numbers. Divide, keep the remainder, and repeat with the divisor and the remainder until the remainder is 0; the last non-zero remainder is the GCD:

180=3·48+36, then 48=1·36+12, then 36=3·12+0. So \gcd(48,180)=12.

It works because any number dividing a and b also divides a-qb, and the other way round, so \gcd(a,b)=\gcd(b, a-qb) while the numbers shrink to zero. Euclid wrote it down in Book VII of the Elements, around 300 BCE, which makes it one of the oldest algorithms still in daily use.

The two are linked: \gcd(a,b)·\operatornamelcm(a,b)=a· b. Check: 12·720=8640=48·180. So \operatornamelcm(a,b)=(a· b)/(\gcd(a,b)). Two numbers are coprime when their GCD is 1, as with 8 and 15.

What they are for.

• Simplifying a fraction: divide top and bottom by the GCD. (48)/(180)=(48÷12)/(180÷12)=(4)/(15).
• Cutting into equal pieces: a floor 48 dm by 180 dm is tiled with the largest square tiles whose side is \gcd=12 dm, which takes 4·15=60 tiles.
• Repeating events: buses leave every 12 and every 18 minutes, together at 7:00. They next leave together after \operatornamelcm(12,18)=36 minutes.
• Adding fractions: the least common denominator is the LCM of the denominators.

What is division with remainder and modular arithmetic?

Dividing an integer a by b≠0 gives a unique quotient q and remainder r with a=b· q+r and 0\le r<|b|; the remainder is written a\bmod b. For example 17=5·3+2, so 17\bmod5=2.

With a negative dividend the rule still holds, and the remainder stays between 0 and |b|-1: -7=3·(-3)+2, so -7\bmod3=2, not -1. Think of a clock: 7 hours before 12 o'clock is 5 o'clock, and a day of 24 hours repeats after 24, which is the same as counting remainders.

Modular arithmetic works with remainders only. Two integers are congruent modulo n, written a\equiv c\pmod n, when n divides a-c, that is, when they leave the same remainder: 17\equiv2\pmod5. Remainders respect addition and multiplication, so you can reduce first and calculate afterward:

• Days of the week. 100 days after a Monday: 100\bmod7=2, so it is a Wednesday.
• Last digits. The last digit of 7^k repeats 7, 9, 3, 1. Since 2026\bmod4=2, the last digit of 7^2026 is the second in the cycle, 9.
• Parity. Even and odd are just a\bmod2.
• Check digits and hashing. Barcodes and card numbers carry a digit computed with \bmod10 to catch typing errors, and a hash table picks a slot with \bmod the table size.

The catch is that programming languages do not all agree on negative operands. Each satisfies a=b· q+r but chooses q differently: mathematics keeps r\ge0; Python rounds q down, so r takes the sign of the divisor; JavaScript, C, C++ and Java truncate q toward zero, so r takes the sign of the dividend. Compare them below.

How do you work with integers in Python and JavaScript?

Python integers have no size limit and use floor division, while JavaScript numbers are floating-point numbers (floats), exact only up to 2^53-1, with a remainder that follows the dividend; use BigInt in JavaScript for large exact integers. The two languages disagree exactly where this article has been pointing. Why fixed-size integers wrap around is explained by two’s complement in the article on binary numbers.

>>> 7 // 2, -7 // 2          # floor division rounds down
(3, -4)
>>> 7 % 3, -7 % 3, 7 % -3        # the remainder takes the sign of the divisor
(1, 2, -2)
>>> divmod(-7, 3)                # quotient and remainder together
(-3, 2)
>>> int(-7 / 2)                  # / gives a float, int() truncates toward zero
-3
>>> import math
>>> math.gcd(48, 180), math.lcm(4, 6)
(12, 12)
>>> 2 ** 100                     # no overflow
1267650600228229401496703205376

def gcd(a, b):
    while b:
        a, b = b, a % b
    return abs(a)

def factorize(n):
    out, p = [], 2
    while p * p <= n:
        while n % p == 0:
            out.append(p)
            n //= p
        p += 1
    if n > 1:
        out.append(n)
    return out

gcd(48, 180)      # 12
factorize(360)    # [2, 2, 2, 3, 3, 5]

-7 % 3                          // -1   the remainder takes the sign of the dividend
((-7 % 3) + 3) % 3              // 2    the usual fix for a non-negative remainder
Math.trunc(-7 / 2)              // -3
Math.floor(-7 / 2)              // -4
Number.MAX_SAFE_INTEGER         // 9007199254740991
2 ** 53 + 1 === 2 ** 53         // true: beyond 2^53 integers lose exactness
2n ** 100n                      // 1267650600228229401496703205376n
-7n % 3n                        // -1n  BigInt behaves like Number here
7n / 2n                         // 3n   BigInt division truncates

#include <iostream>
using namespace std;

int mod(int a, int b) { return ((a % b) + b) % b; }       // a non-negative remainder, for b > 0
int floorDiv(int a, int b) { return (a - mod(a, b)) / b; }
int gcd(int a, int b) {                                    // Euclid's algorithm
  while (b != 0) { int t = a % b; a = b; b = t; }
  return a < 0 ? -a : a;
}
int lcm(int a, int b) { return a / gcd(a, b) * b; }

int main() {
  cout << 7 / 2 << " " << 7 % 3 << endl;                   // 3 1
  // C++ truncates toward zero: -7 / 2 is -3 and -7 % 3 is -1
  cout << floorDiv(-7, 2) << " " << mod(-7, 3) << endl;    // floor division and the usual remainder
  cout << gcd(48, 180) << " " << lcm(4, 6) << endl;
  int n = 360;                                             // factorization by trial division
  for (int p = 2; p * p <= n; p++)
    while (n % p == 0) { cout << p << " "; n /= p; }
  if (n > 1) cout << n;
  cout << endl;
  return 0;
}

/* output:
3 1
-4 2
12 12
2 2 2 3 3 5
*/

The pitfalls, in order of how often they bite:

Python's math.gcd and math.lcm (from Python 3.9) take any size of integer, so you rarely need your own, but writing Euclid's loop once is the best way to see why it ends.

What are the common mistakes with integers?

The most common mistakes with integers are the nine below, each with the correct statement and a number that shows why.

Practice: test your understanding

These questions mix everything above. A wrong answer costs nothing here: read the hint and try again.

Summary: integers at a glance

• Integers: …,-2,-1,0,1,2,…; zero is neither positive nor negative; every a has an opposite -a.
• Order and size: further right is greater, so -7<-3; |a| is the distance from 0.
• Adding and subtracting: same signs add and keep the sign; different signs subtract and keep the sign of the larger; a-b=a+(-b).
• Multiplying and dividing: same signs give positive, different signs give negative; never divide by 0.
• Order of operations: brackets, powers, then ×÷, then +− left to right; -3^2=-9 but (-3)^2=9.
• Divisibility: digit rules for 2, 3, 4, 5, 6, 8, 9, 10, 11; every integer above 1 has one prime factorization.
• GCD and LCM: smaller or larger prime exponents, or Euclid's algorithm; \gcd·\operatornamelcm=a· b.
• Remainders: a=bq+r with 0\le r<|b|; Python, JavaScript and mathematics differ for negatives.

Glossary: key terms

Integer

A number in the set ℤ, with no fractional part: zero, the positive whole numbers and their negatives.

Opposite

The number on the other side of zero at the same distance, so that a number plus its opposite is zero.

Absolute value

The distance of a number from zero, which is never negative, so the absolute value of minus 7 is 7.

Closed under an operation

A set is closed under an operation when applying it to members of the set always gives a member; the integers are closed under addition, subtraction and multiplication.

Divisor (factor)

An integer b is a divisor of a when a equals b times some integer, so that division leaves no remainder.

Multiple

The product of a number and any integer, such as 12, 24 and 36 for the number 12.

Prime number

An integer greater than 1 whose only positive divisors are 1 and itself.

Composite number

An integer greater than 1 that is not prime, so it has a divisor other than 1 and itself.

Prime factorization

The unique way of writing an integer greater than 1 as a product of primes, such as 360 equal to 2 cubed times 3 squared times 5.

Greatest common divisor (GCD)

The largest integer that divides both of two given integers; for 48 and 180 it is 12.

Least common multiple (LCM)

The smallest positive integer that is a multiple of both of two given integers; for 12 and 18 it is 36.

Coprime

Two integers whose greatest common divisor is 1, such as 8 and 15.

Remainder

What is left after dividing: in a equal to b times q plus r, the number r, which lies from 0 up to the size of b minus 1.

Modulo

The operation that gives the remainder of a division, so that 17 modulo 5 is 2.

Frequently asked questions

References

1. James Stewart, Lothar Redlin and Saleem Watson (2016). Precalculus: Mathematics for Calculus (7th ed.). Cengage Learning.
2. Euclid. Elements, Book VII (the Euclidean algorithm) and Book IX, proposition 20 (there are infinitely many primes). c. 300 BCE.
3. Brahmagupta (628). Brahmasphutasiddhanta (rules for arithmetic with zero, debts and fortunes).
4. Carl Friedrich Gauss (1801). Disquisitiones Arithmeticae.
5. Ronald Graham, Donald Knuth and Oren Patashnik (1994). Concrete Mathematics: A Foundation for Computer Science (2nd ed.), chapter 3 (floor, ceiling and mod). Addison-Wesley.
6. Shen Kangshen, John Crossley and Anthony Lun (1999). The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford University Press.
7. Python Software Foundation. The Python Language Reference: binary arithmetic operations (floor division and modulo). docs.python.org. https://docs.python.org/3/reference/expressions.html#binary-arithmetic-operations.
8. Ecma International. ECMAScript Language Specification: multiplicative operators (the remainder operator). tc39.es. https://tc39.es/ecma262/#sec-multiplicative-operators.

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