Modulo Calculator (a mod n, Congruence and Modular Exponentiation)

Choose a mode and enter integers. "Remainder" also works with negative numbers and shows how the math remainder (0 or more) differs from the % operator in programming.

What to calculate

Integer a

Integer b (to compare)

Exponent k (power)

Modulus n (divisor)

Enter whole numbers only (no decimals or fractions). a, b and n can have up to 30 digits, the exponent k is 0 or more with up to 9 digits, and the modulus n is 1 or more.

Result and figure

Choose a mode on the left, enter integers and press "Calculate". The result will appear here (when the modulus n is 2 to 24, a clock diagram also shows where the remainder lands).

[ CLOSE ]

What you can do on this page
What is this calculation used for?
How to Use
Formulas and figures
Symbols and terms
- Symbols
- Terms
Good to know before you start
How to calculate it in Excel
How to calculate it in Google Sheets
How to calculate it in Python
How to write it in LaTeX and other math languages (copy and paste)
How to have ChatGPT do the calculation
DataChef Features
Related Features
NumberChef Calculators List

What you can do on this page

Find the remainder when an integer $a$ is divided by $n$. Negative numbers work too (for example, $-7$ divided by $3$): the page shows the math remainder ($0 \le r < n$) side by side with the result of the % operator in C, Java, JavaScript and similar languages (which can be negative)
Check whether the congruence $a \equiv b \pmod{n}$ holds, with the reason: whether the difference $a - b$ is a multiple of $n$
Get the exact remainder of a huge power, such as $7^{100}$ divided by $13$, with the steps of repeated squaring (numbers too big for a regular calculator are no problem)
See the remainder on a clock diagram (the circle of mod $n$) and get a feel for how remainders go around and around the same positions (the clock appears when the modulus $n$ is 2 to 24)
A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page

The modulus $n$ is an integer of 1 or more. Enter $a$, $b$ and $n$ with up to 30 digits, and the exponent $k$ as an integer of 0 or more with up to 9 digits.

What is this calculation used for?

Clock and calendar math (what day of the week is it in N days?)

On a clock, "14:00 is 2 p.m." is exactly the remainder calculation 14 mod 12 = 2. Days of the week work the same way, on a clock that goes around once every 7 days. For example, if today is Monday, then since 100 mod 7 = 2, the day 100 days from now is "2 days after Monday", which is Wednesday.
Paper planners and calendar apps alike match dates to days of the week with this remainder calculation.

Check digits that catch wrong numbers (barcodes and ISBNs)

The last digit of a product barcode (UPC) or a book's ISBN is a check digit made from the other digits with a remainder calculation. For a UPC, for example, the digits are multiplied alternately by 3 and 1 and added up, and the last digit is chosen from the remainder of that total divided by 10.
If one digit is mistyped, the remainder no longer matches, so checkout registers and online stores can catch the mistake on the spot. The check digit in credit card numbers works on the same idea.

Encryption that protects the internet (RSA)

RSA, the best-known public-key encryption for sending things like credit card numbers safely, uses "the remainder of a huge power" directly in its calculations. A power mod can be computed quickly, but working backward from the result to the original number is extremely hard. This one-way property is what makes it secure.
Repeated squaring, whose steps the "Power mod" mode on this page shows, is the calculation that encryption software runs every day.

Everyday programming (even or odd, taking turns in a loop)

In programming, the % operator is used all the time: i % 2 tells even from odd, and i % n makes an order that loops around every n items (choosing the next player in a turn-based game, going back to the first slide after the last one, and so on).
When negative numbers are involved, % gives different results in different languages, so knowing "the math remainder versus % in C and Java" helps when you hunt for bugs.

Sharing equally and counting what is left (handing out, packing)

Share 63 pencils equally among 12 people, and each gets 5 with 3 left over (63 = 12 × 5 + 3). Packing stock into boxes (how many full boxes of n, and how many left over), splitting people into groups for a party: every "share equally and see what is left" situation is this calculation.
The basic division equation a = n × q + r puts all these everyday situations into one formula.

Formulas and figures

The basic division equation (what a remainder is)

Clock diagram (mod 12)

Standard notation (the usual math form)

$a$ $=$ $n$ $\times$ $q$ $+$ $r$

In words (symbols replaced with words)

④ $a$: dividend $=$ ① $n$: divisor (modulus) $\times$ ② $q$: quotient $+$ ③ $r$: remainder

The formula in words

① Take the $n$: divisor (modulus)

② multiply it by the $q$: quotient

③ and add the $r$: remainder (chosen so that $0 \le r < n$)

④ and you get back the $a$: dividend . There is exactly one way to write it in this form, and $r$ is "the remainder of $a$ divided by $n$" ($a \bmod n$)

Quick example

Divide 38 by 12 (the same calculation that turns "hour 38" into 2 o'clock on a clock)

dividend 38 $=$ divisor 12 $\times$ quotient 3 $+$ remainder 2

$38 = 12 \times 3 + 2$

$38 \bmod 12 = 2$

Key idea

In math, the remainder $r$ is always chosen in the range $0 \le r < n$. The same goes for negative numbers. To divide $-7$ by $3$, write $-7 = 3 \times (-3) + 2$: the quotient is lowered to the smaller integer ($-3$) so that the remainder is $2$ (floor division). The % operator in programming, however, works differently from language to language. In C, Java and JavaScript, -7 % 3 rounds the quotient toward 0 (to $-2$), so the result is $-1$ (the remainder takes the sign of the dividend). The % operator in Python and Ruby and the MOD function in Excel give the remainder the sign of the divisor, so they return $2$, the same as math. In programs that handle remainders of negative numbers, this difference is a common source of bugs, so be careful.

Definition of congruence (what a ≡ b (mod n) says)

Standard notation (the usual math form)

$a - b$ $=$ $n$ $\times$ $m$

In words (symbols replaced with words)

③ $a - b$: difference of the two integers $=$ ① $n$: modulus $\times$ ② $m$: some integer

The formula in words

① When the $n$: modulus

② times $m$: some integer

③ is exactly equal to the $a - b$: difference of the two integers (that is, when the difference is a multiple of $n$), you write $a \equiv b \pmod{n}$ and say "$a$ and $b$ are congruent modulo $n$"

Quick example

38 and 14 are congruent modulo 12 ($38 \equiv 14 \pmod{12}$)

difference (38 − 14 = 24) $=$ modulus 12 $\times$ integer 2

$38 - 14 = 24 = 12 \times 2$

$38 \equiv 14 \pmod{12}$

Key idea

"The difference is a multiple of $n$" and "the remainders when divided by $n$ are equal" say the same thing in two ways (indeed $38 \bmod 12 = 2$ and $14 \bmod 12 = 2$, so the remainders match). That is why you can read a congruence as "an equation that looks only at remainders". Picture a clock: hour 38, hour 14 and hour 2 all point to the same spot on the dial. The world of mod $n$ is a clock dial that goes around once every $n$, and congruent numbers are numbers that land at the same spot on that dial.

Multiplication and remainders (the basis of power mod)

Standard notation (the usual math form)

$(a \times b) \bmod n$ $=$ $\{(a \bmod n) \times (b \bmod n)\} \bmod n$

In words (symbols replaced with words)

② remainder of the product $=$ ① remainder after multiplying the two remainders

The formula in words

① The remainder after multiplying the two remainders

② is the same as the remainder of the product (you may turn each number into its remainder first and then multiply)

Quick example

The remainder of 38 × 15 divided by 12 can be found from the two remainders, 2 and 3, alone

remainder of the product (38 × 15 = 570) $=$ remainder 2 × remainder 3 = 6

$38 \bmod 12 = 2,\quad 15 \bmod 12 = 3$

$2 \times 3 = 6$

$(38 \times 15) \bmod 12 = 570 \bmod 12 = 6$

Key idea

Thanks to this property, you never have to finish a big multiplication: you can turn the numbers into remainders first and multiply those. A power is repeated multiplication, so you can find the remainder of something like $7^{100}$ by following only the remainders, without ever building the huge number. This is the basis of repeated squaring (square over and over, taking the remainder each time, split the exponent into a sum of powers of 2 and multiply the pieces). The "Power mod" mode of this calculator shows its steps this way. Also, a power has only $n$ possible remainders, so it must come back to a remainder it had before and then repeat in a cycle. For example, the last digit of $7^{k}$ (its remainder mod $10$) goes around 7, 9, 3, 1 in turn. Finding such a cycle is a classic technique in math contests.

Properties of congruences (adding, subtracting and multiplying both sides)

Standard notation (the usual math form)

$a + c$ $\equiv$ $b + d$ $\pmod{n}$

$a - c$ $\equiv$ $b - d$ $\pmod{n}$

$a \times c$ $\equiv$ $b \times d$ $\pmod{n}$

In words (symbols replaced with words)

① $a + c$: sum of the left sides $\equiv$ ② $b + d$: sum of the right sides $\pmod{n}$

$a - c$: difference of the left sides $\equiv$ $b - d$: difference of the right sides $\pmod{n}$

$a \times c$: product of the left sides $\equiv$ $b \times d$: product of the right sides $\pmod{n}$

The formula in words

① If $a \equiv b$ and $c \equiv d \pmod{n}$, then the sum, difference or product of the left sides

② stays congruent to the sum, difference or product of the right sides (you can add, subtract and multiply congruences just like ordinary equations)

Quick example

Using $38 \equiv 2$ and $15 \equiv 3 \pmod{12}$:

38 + 15 = 53 $\equiv$ 2 + 3 = 5 $\pmod{12}$

$38 + 15 = 53 \equiv 5,\quad 2 + 3 = 5 \pmod{12}$

$38 \times 15 = 570 \equiv 6,\quad 2 \times 3 = 6 \pmod{12}$

Key idea

What makes this property handy is that even with big numbers, you can turn them into remainders first and then add, subtract or multiply. Division is the one thing you cannot do freely. For example, $6 \equiv 12 \pmod{6}$ is true, but dividing both sides by 2 gives $3 \equiv 6 \pmod{6}$, which is false (the difference 3 is not a multiple of 6). You may divide both sides only when the number you divide by and the modulus are relatively prime (their greatest common divisor is 1).

The remainder of an integer $a$ divided by $n$ is the $r$ in the one and only way to write $a = n \times q + r$ with $0 \le r < n$. The congruence $a \equiv b \pmod{n}$ says that the difference of $a$ and $b$ is a multiple of $n$ (in other words, they have the same remainder when divided by $n$), and you can add, subtract and multiply congruences just like equations. Thanks to this, even the remainder of a huge power can be found quickly by repeated squaring, which follows only the remainders.

Symbols and terms

Symbols

$\equiv$	is congruent to	The symbol for congruence. It is an equals sign with three lines instead of two, and says "equal in the world of remainders", which is looser than "exactly equal".
$\bmod$ (mod)	mod	Short for "modulus", from the Latin word modulus (a small measure). "$a \bmod n$" stands for "the remainder of $a$ divided by $n$", and "$\pmod{n}$" written after an equation declares "we are working modulo $n$".
$a,\ b$	a, b	The integers whose remainders you look at. In a congruence, they are the two integers on the left and right. Letters near the start of the alphabet are traditionally used for fixed numbers.
$n$	n	The modulus (the number you divide by). The letter $n$, from "number", is often used. On this page it is an integer of 1 or more.
$q$	q	The quotient, from the first letter of "quotient". When dividing a negative number, the quotient is rounded down to the smaller integer so that the remainder is 0 or more (floor division).
$r$	r	The remainder, from the first letter of "remainder". In math it is always chosen in the range $0 \le r < n$.
$m$	m	The integer in the definition of congruence that tells how many times the modulus goes into the difference. It can be negative or 0.
$a^{k}$	a to the k	$a$ multiplied by itself $k$ times (a power). The small raised number $k$ is the exponent, which tells how many times to multiply.
%	percent (as an operator, mod)	The modulo operator that finds a remainder in many programming languages. It is the same sign as the percent sign but has a different job. In C, Java and JavaScript, its result for negative numbers can differ from the math remainder (see the key idea of formula card 1).

Terms

remainder (residue)	What is left over when a division does not come out even. "Residue" is a more formal word for the same thing. In math it is always at least $0$ and less than the divisor.
quotient	The integer that tells how many times the divisor can be taken away. It is the $q$ in the basic division equation $a = n \times q + r$.
modulus	The number $n$ you divide by to get the remainders you work with. On a clock dial, it is the number of marks in one full turn. "Modulo 12" is short for "looking at remainders when divided by 12".
congruent	Two integers are congruent modulo $n$ when they have the same remainder when divided by $n$. This is a term about integers, different from congruent shapes in geometry (same shape and size).
congruence	A statement of the form $a \equiv b \pmod{n}$. You can add, subtract and multiply congruences just like equations, so remainder problems can be worked out by rewriting. It is a basic tool of number theory, used in math contests and computer science.
multiple	A number you get by multiplying an integer by another integer. "The difference is a multiple of the modulus" is the definition of congruence.
floor division	A way of dividing that rounds the quotient down to the smaller integer. When the divisor is positive, the remainder is 0 or more even for a negative dividend. The // operator in Python and the INT function in Excel work this way.
modulo operator	The % operator that finds a remainder in programming. C, Java and JavaScript round the quotient toward 0 (the remainder takes the sign of the dividend). Python and Ruby use floor division (the remainder takes the sign of the divisor, so it is 0 or more when the modulus is positive).
repeated squaring	A way to find the remainder of a huge power with few multiplications: square over and over while taking the remainder, split the exponent into a sum of powers of 2 and multiply the pieces. It is also called exponentiation by squaring or square-and-multiply, and is used in implementations of RSA encryption and more.
residue class	The idea of grouping integers by their remainder when divided by $n$. In the world of mod $n$, every integer falls into one of $n$ groups, remainders $0$ to $n-1$. Even and odd numbers are the grouping by remainder when divided by 2.
periodicity	The way the remainders of powers repeat the same pattern. There are only $n$ possible remainders, so they must come back to one they had before. Finding the cycle in last-digit problems (mod 10) is a classic in math contests.
relatively prime	Two integers are relatively prime (or coprime) when their greatest common divisor is 1. It appears in the condition for dividing both sides of a congruence by the same number.

Good to know before you start

Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back to these topics is the quickest way forward.

Division with remainders (Grades 3–4)	Being able to calculate $17 \div 5 = 3$ remainder $2$ Being able to check it with "divisor × quotient + remainder = dividend"
Multiples and factors (Grades 4–6)	Knowing that $24$ is a multiple of $12$ Knowing what the greatest common factor is (it is used in the condition "relatively prime" for dividing a congruence)
Negative numbers (Grades 6–7)	Being able to add, subtract and multiply with negative numbers (for example, $3 \times (-3) = -9$) Knowing that $-3$ is less than $-2$ (to its left) on the number line
Exponents (Grades 6–8)	Knowing that the raised exponent tells how many times to multiply, as in $3^{4} = 3 \times 3 \times 3 \times 3 = 81$ Being able to use the laws of exponents $a^{m} \times a^{n} = a^{m+n}$ and $(a^{m})^{n} = a^{mn}$ (they are why repeated squaring works)
The division algorithm for integers (high school)	Knowing that dividing an integer $a$ by a positive integer $n$ can be written as $a = nq + r$ (with $0 \le r < n$) in exactly one way Having met the idea of grouping integers by their remainders (even and odd numbers are the grouping by remainder when divided by 2)

How to calculate it in Excel

Copy the whole table below and paste it into cell A1 in Excel. It works as is.

Table to find a mod n (the remainder)

Dividend a	-7
Modulus (divisor) n	3
Math remainder (0 or more)	=MOD(B1,B2)
Quotient (floor division)	=INT(B1/B2)
Check n × quotient + remainder	=B2*B4+B3
C/Java-style remainder (rounded toward 0)	=B1-B2*TRUNC(B1/B2)

Table to check a ≡ b (mod n)

Integer a	38
Integer b	14
Modulus n	12
Difference a − b	=B1-B2
Remainder of the difference ÷ n	=MOD(B4,B3)
Result (TRUE = congruent)	=MOD(B4,B3)=0

Table to find a power mod n (repeated squaring)

Base a	7
Exponent k	100
Modulus n	13
Remainder of a^1	=MOD(B1,B3)
Remainder of a^2 (square the row above, take the remainder)	=MOD(B4^2,$B$3)
Remainder of a^4	=MOD(B5^2,$B$3)
Remainder of a^8	=MOD(B6^2,$B$3)
Remainder of a^16	=MOD(B7^2,$B$3)
Remainder of a^32	=MOD(B8^2,$B$3)
Remainder of a^64	=MOD(B9^2,$B$3)
Remainder combined for 100 = 64 + 32 + 4	=MOD(MOD(B10B9,$B$3)B6,$B$3)

After pasting, just change the upper rows (the inputs), and the lower rows are calculated automatically.
Excel's MOD function returns the same "math remainder" (0 or more) as this calculator. Even for negative numbers, =MOD(-7,3) is 2. If you need the C/Java-style remainder, use the last row of the first table, which uses TRUNC to round the quotient toward 0.
The first table divides −7 by 3: the remainder is 2 and the quotient is −3.
The second table checks 38 ≡ 14 (mod 12). The remainder of the difference 24 is 0, so it shows TRUE (congruent).
The third table finds the remainder of 7 to the 100th power divided by 13 by repeated squaring. The answer is 9. "^" is the power sign. Excel handles only about 15 digits exactly, so when the modulus n has more than 7 digits, the squared values lose precision. In that case, use the calculator on this page or Python.

How to calculate it in Google Sheets

Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.

Table to find a mod n (the remainder)

Dividend a	-7
Modulus (divisor) n	3
Math remainder (0 or more)	=MOD(B1,B2)
Quotient (floor division)	=INT(B1/B2)
Check n × quotient + remainder	=B2*B4+B3
C/Java-style remainder (rounded toward 0)	=B1-B2*TRUNC(B1/B2)

Table to check a ≡ b (mod n)

Integer a	38
Integer b	14
Modulus n	12
Difference a − b	=B1-B2
Remainder of the difference ÷ n	=MOD(B4,B3)
Result (TRUE = congruent)	=MOD(B4,B3)=0

Table to find a power mod n (repeated squaring)

Base a	7
Exponent k	100
Modulus n	13
Remainder of a^1	=MOD(B1,B3)
Remainder of a^2 (square the row above, take the remainder)	=MOD(B4^2,$B$3)
Remainder of a^4	=MOD(B5^2,$B$3)
Remainder of a^8	=MOD(B6^2,$B$3)
Remainder of a^16	=MOD(B7^2,$B$3)
Remainder of a^32	=MOD(B8^2,$B$3)
Remainder of a^64	=MOD(B9^2,$B$3)
Remainder combined for 100 = 64 + 32 + 4	=MOD(MOD(B10B9,$B$3)B6,$B$3)

The same formulas as in Excel (MOD, INT, TRUNC) work as is. Copy the whole table, paste it into cell A1, and replace the inputs with your own numbers.

How to calculate it in Python

a = -7
n = 3

# Python's % returns the same "math remainder" as this calculator (0 or more when the modulus is positive)
print(a % n)                 # 2

# divmod returns the floor-division quotient and the remainder together (a = n × quotient + remainder)
quotient, remainder = divmod(a, n)
print(quotient, remainder)   # -3 2

# congruence check: is 38 ≡ 14 (mod 12)? (check whether the difference is a multiple of 12)
print((38 - 14) % 12 == 0)   # True

# power mod: remainder of 7 to the 100th power divided by 13
# the 3-argument pow uses repeated squaring, so it finishes instantly even for huge exponents
print(pow(7, 100, 13))       # 9

def c_style_mod(x, m):
    # same result as % in C, Java and JavaScript (quotient rounded toward 0)
    r = x % m
    if r != 0 and x < 0:
        r -= m
    return r

print(c_style_mod(-7, 3))    # -1

Python's % operator matches the math remainder. When the divisor is positive, it returns a value of 0 or more even for negative numbers (-7 % 3 is 2). For power mod, the 3-argument form pow(base, exponent, modulus) runs repeated squaring inside, so it finds just the remainder quickly even for powers with tens of thousands of digits. When you need the same result as % in C, Java and JavaScript, add a correction that rounds the quotient toward 0, as in the last function.

How to write it in LaTeX and other math languages (copy and paste)

The basic division equation (what a remainder is)

a = n × q + r (0 ≤ r < n)

a = nq + r \quad (0 \le r < n)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>a</mi><mo>=</mo><mi>n</mi><mi>q</mi><mo>+</mo><mi>r</mi>
    <mo>,</mo>
    <mn>0</mn><mo>&#x2264;</mo><mi>r</mi><mo>&lt;</mo><mi>n</mi>
  </mrow>
</math>

a = n q + r, \ 0 <= r < n

{Quotient[a, n], Mod[a, n]}

q := floor(a/n); r := a mod n;

q = floor(a/n); r = mod(a, n);

a = nq + r (0 ≤ r < n)

Definition of congruence (what a ≡ b (mod n) says)

a ≡ b (mod n)  ⇔  a − b = n × m

a \equiv b \pmod{n} \iff a - b = nm

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>a</mi><mo>&#x2261;</mo><mi>b</mi>
    <mspace width="0.3em"/>
    <mo>(</mo><mi>mod</mi><mspace width="0.3em"/><mi>n</mi><mo>)</mo>
    <mo>&#x21D4;</mo>
    <mi>a</mi><mo>&#x2212;</mo><mi>b</mi><mo>=</mo><mi>n</mi><mi>m</mi>
  </mrow>
</math>

a -= b (mod n) iff a - b = n m

Mod[a - b, n] == 0

(a - b) mod n = 0;

mod(a - b, n) == 0

a ≡ b (mod n)

Multiplication and remainders (the basis of power mod)

(a × b) mod n = {(a mod n) × (b mod n)} mod n

(a \times b) \bmod n = \{(a \bmod n)(b \bmod n)\} \bmod n

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mo>(</mo><mi>a</mi><mo>&#xD7;</mo><mi>b</mi><mo>)</mo>
    <mspace width="0.3em"/><mi>mod</mi><mspace width="0.3em"/><mi>n</mi>
    <mo>=</mo>
    <mo>{</mo>
    <mo>(</mo><mi>a</mi><mspace width="0.3em"/><mi>mod</mi><mspace width="0.3em"/><mi>n</mi><mo>)</mo>
    <mo>&#xD7;</mo>
    <mo>(</mo><mi>b</mi><mspace width="0.3em"/><mi>mod</mi><mspace width="0.3em"/><mi>n</mi><mo>)</mo>
    <mo>}</mo>
    <mspace width="0.3em"/><mi>mod</mi><mspace width="0.3em"/><mi>n</mi>
  </mrow>
</math>

(a * b) mod n = ((a mod n) * (b mod n)) mod n

Mod[a b, n] == Mod[Mod[a, n] Mod[b, n], n]

(a * b) mod n = ((a mod n) * (b mod n)) mod n;

mod(a*b, n) == mod(mod(a, n)*mod(b, n), n)

(a×b) mod n = ((a mod n)×(b mod n)) mod n

Properties of congruences (adding, subtracting and multiplying both sides)

a ≡ b, c ≡ d (mod n) ⇒ a+c ≡ b+d, a−c ≡ b−d, a×c ≡ b×d (mod n)

a \equiv b,\ c \equiv d \pmod{n} \Rightarrow a + c \equiv b + d,\ a - c \equiv b - d,\ ac \equiv bd \pmod{n}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>a</mi><mo>+</mo><mi>c</mi><mo>&#x2261;</mo><mi>b</mi><mo>+</mo><mi>d</mi>
    <mo>,</mo>
    <mi>a</mi><mo>&#x2212;</mo><mi>c</mi><mo>&#x2261;</mo><mi>b</mi><mo>&#x2212;</mo><mi>d</mi>
    <mo>,</mo>
    <mi>a</mi><mi>c</mi><mo>&#x2261;</mo><mi>b</mi><mi>d</mi>
    <mspace width="0.3em"/>
    <mo>(</mo><mi>mod</mi><mspace width="0.3em"/><mi>n</mi><mo>)</mo>
  </mrow>
</math>

a + c -= b + d, \ a - c -= b - d, \ a c -= b d (mod n)

Mod[a + c, n] == Mod[b + d, n] && Mod[a - c, n] == Mod[b - d, n] && Mod[a c, n] == Mod[b d, n]

(a + c) mod n = (b + d) mod n;

mod(a + c, n) == mod(b + d, n)

a + c ≡ b + d (mod n)

How to have ChatGPT do the calculation

You are a math calculation assistant for number theory (properties of integers). Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing).

Calculate the following 3 problems.
1. The math remainder of −7 divided by 3 (chosen to be 0 or more), and the quotient
2. Whether 38 ≡ 14 (mod 12) holds (also show whether the difference is a multiple of 12)
3. The remainder of 7 to the 100th power divided by 13 (use pow(7, 100, 13))

Use Python's %, divmod and pow(base, exponent, modulus), and show the formulas you used and the numbers from the execution result.

How to Use

1

Enter your numbers

Type the numbers you want to calculate with into the input fields
2

Calculate

Press the "Calculate" button
3

Check the result

The result appears on the spot. The same page also explains the idea behind the calculation and the formula

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\(\equiv\)	is congruent to	The symbol for congruence. It is an equals sign with three lines instead of two, and says "equal in the world of remainders", which is looser than "exactly equal".
\(\bmod\) (mod)	mod	Short for "modulus", from the Latin word modulus (a small measure). "\(a \bmod n\)" stands for "the remainder of \(a\) divided by \(n\)", and "\(\pmod{n}\)" written after an equation declares "we are working modulo \(n\)".
\(a,\ b\)	a, b	The integers whose remainders you look at. In a congruence, they are the two integers on the left and right. Letters near the start of the alphabet are traditionally used for fixed numbers.
\(n\)	n	The modulus (the number you divide by). The letter \(n\), from "number", is often used. On this page it is an integer of 1 or more.
\(q\)	q	The quotient, from the first letter of "quotient". When dividing a negative number, the quotient is rounded down to the smaller integer so that the remainder is 0 or more (floor division).
\(r\)	r	The remainder, from the first letter of "remainder". In math it is always chosen in the range \(0 \le r < n\).
\(m\)	m	The integer in the definition of congruence that tells how many times the modulus goes into the difference. It can be negative or 0.
\(a^{k}\)	a to the k	\(a\) multiplied by itself \(k\) times (a power). The small raised number \(k\) is the exponent, which tells how many times to multiply.
%	percent (as an operator, mod)	The modulo operator that finds a remainder in many programming languages. It is the same sign as the percent sign but has a different job. In C, Java and JavaScript, its result for negative numbers can differ from the math remainder (see the key idea of formula card 1).

Division with remainders (Grades 3–4)	Being able to calculate \(17 \div 5 = 3\) remainder \(2\) Being able to check it with "divisor × quotient + remainder = dividend"
Multiples and factors (Grades 4–6)	Knowing that \(24\) is a multiple of \(12\) Knowing what the greatest common factor is (it is used in the condition "relatively prime" for dividing a congruence)
Negative numbers (Grades 6–7)	Being able to add, subtract and multiply with negative numbers (for example, \(3 \times (-3) = -9\)) Knowing that \(-3\) is less than \(-2\) (to its left) on the number line
Exponents (Grades 6–8)	Knowing that the raised exponent tells how many times to multiply, as in \(3^{4} = 3 \times 3 \times 3 \times 3 = 81\) Being able to use the laws of exponents \(a^{m} \times a^{n} = a^{m+n}\) and \((a^{m})^{n} = a^{mn}\) (they are why repeated squaring works)
The division algorithm for integers (high school)	Knowing that dividing an integer \(a\) by a positive integer \(n\) can be written as \(a = nq + r\) (with \(0 \le r < n\)) in exactly one way Having met the idea of grouping integers by their remainders (even and odd numbers are the grouping by remainder when divided by 2)

Modulo Calculator (a mod n, Congruence and Modular Exponentiation)

Table of Contents

What you can do on this page

What is this calculation used for?

Formulas and figures

Symbols and terms

Symbols

Terms

Good to know before you start

How to calculate it in Excel

How to calculate it in Google Sheets

How to calculate it in Python

How to write it in LaTeX and other math languages (copy and paste)

How to have ChatGPT do the calculation

How to Use

DataChef Features

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