The ModuloSteps command accepts an expression that is expected to contain modulos and displays the steps required to evaluate each modulo given.

If expr is a string, then it is parsed into an expression using InertForm:-Parse so that no automatic simplifications are applied, and thus no steps are missed.  

The implicitmultiply option is only relevant when expr is a string.  This option is passed directly on to the InertForm:-Parse command and will cause things like 2x to be interpreted as 2*x, but also, xyz to be interpreted as x*y*z.

The output and displaystyle options are described in Student:-Basics:-OutputStepsRecord. The return value is controlled by the output option.  

This function is part of the Student:-Basics package.

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{Basics}\right)\:$

$\mathrm{ModuloSteps}\left(''4\; mod\; 7''\right)$

$\begin{array}{lll}& & 4\mathbf{mod}7\\ \text{•}& & \text{Evaluate modulo, which is the remainder of}4\text{divided by}7\\ & & 4\end{array}$

$\mathrm{ModuloSteps}\left(''((3\; +\; 5\; +\; 6)\; mod\; 4)\; +\; ((5*6*7)\; mod\; 3)''\right)$

$\begin{array}{lll}& & \left(\left(3+5+6\right)\mathbf{mod}4\right)+\left(\left(5\cdot 6\cdot 7\right)\mathbf{mod}3\right)\\ \text{•}& & \text{Examine term}\\ & & \left(3+5+6\right)\mathbf{mod}4\\ \text{•}& & \text{Apply the}\text{addition}\text{mod rule:}\left(a+b\right)\mathbf{mod}m=\left(\left(a\mathbf{mod}m\right)+\left(b\mathbf{mod}m\right)\right)\mathbf{mod}m\\ & & \left(3+\left(5\mathbf{mod}4\right)+\left(6\mathbf{mod}4\right)\right)\mathbf{mod}4\\ \text{•}& & \text{Examine term}\\ & & 5\mathbf{mod}4\\ \text{•}& & \text{Evaluate modulo, which is the remainder of}5\text{divided by}4\\ & & 1\\ \text{•}& & \text{This gives:}\\ & & \left(3+1+\left(6\mathbf{mod}4\right)\right)\mathbf{mod}4\\ \text{•}& & \text{Examine term}\\ & & 6\mathbf{mod}4\\ \text{•}& & \text{Evaluate modulo, which is the remainder of}6\text{divided by}4\\ & & 2\\ \text{•}& & \text{This gives:}\\ & & \left(3+1+2\right)\mathbf{mod}4\\ \text{•}& & \text{Simplify}\\ & & 6\mathbf{mod}4\\ \text{•}& & \text{Evaluate modulo, which is the remainder of}6\text{divided by}4\\ & & 2\\ \text{•}& & \text{This gives:}\\ & & 2+\left(5\cdot 6\cdot 7\mathbf{mod}3\right)\\ \text{•}& & \text{Examine term}\\ & & \left(5\cdot 6\cdot 7\right)\mathbf{mod}3\\ \text{•}& & \text{Apply the}\text{product}\text{mod rule:}\left(a\cdot b\right)\mathbf{mod}m=\left(\left(a\mathbf{mod}m\right)\cdot \left(b\mathbf{mod}m\right)\right)\mathbf{mod}m\\ & & \left(\left(5\mathbf{mod}3\right)\cdot \left(6\mathbf{mod}3\right)\cdot \left(7\mathbf{mod}3\right)\right)\mathbf{mod}3\\ \text{•}& & \text{Examine term}\\ & & 5\mathbf{mod}3\\ \text{•}& & \text{Evaluate modulo, which is the remainder of}5\text{divided by}3\\ & & 2\\ \text{•}& & \text{This gives:}\\ & & \left(2\cdot \left(6\mathbf{mod}3\right)\cdot \left(7\mathbf{mod}3\right)\right)\mathbf{mod}3\\ \text{•}& & \text{Examine term}\\ & & 6\mathbf{mod}3\\ \text{•}& & \text{Evaluate modulo, which is the remainder of}6\text{divided by}3\\ & & 0\\ \text{•}& & \text{This gives:}\\ & & \left(2\cdot 0\cdot \left(7\mathbf{mod}3\right)\right)\mathbf{mod}3\\ \text{•}& & \text{Examine term}\\ & & 7\mathbf{mod}3\\ \text{•}& & \text{Evaluate modulo, which is the remainder of}7\text{divided by}3\\ & & 1\\ \text{•}& & \text{This gives:}\\ & & \left(2\cdot 0\cdot 1\right)\mathbf{mod}3\\ \text{•}& & \text{Simplify}\\ & & 0\mathbf{mod}3\\ \text{•}& & \text{Evaluate modulo, which is the remainder of}0\text{divided by}3\\ & & 0\\ \text{•}& & \text{This gives:}\\ & & 2+0\\ \text{•}& & \text{Simplify}\\ & & 2\end{array}$

$\mathrm{ModuloSteps}\left(''(2^200\; mod\; 24)\; +\; (2^201\; mod\; 24)''\right)$

$\begin{array}{lll}& & \left(\left(2^{200}\right)\mathbf{mod}24\right)+\left(\left(2^{201}\right)\mathbf{mod}24\right)\\ \text{•}& & \text{Examine term}\\ & & \left(2^{200}\right)\mathbf{mod}24\\ \text{•}& & \text{Apply exponent rule:}{a}^{nm+c}={\left(a^n\right)}^m\cdot a^c\\ & & \left({\left(2^7\right)}^{28}\cdot 2^4\right)\mathbf{mod}24\\ \text{•}& & \text{Evaluate inside exponents}\\ & & \left({128}^{28}\cdot 16\right)\mathbf{mod}24\\ \text{•}& & \text{Apply the}\text{product}\text{mod rule:}\left(a\cdot b\right)\mathbf{mod}m=\left(\left(a\mathbf{mod}m\right)\cdot \left(b\mathbf{mod}m\right)\right)\mathbf{mod}m\\ & & \left(\left({128}^{28}\mathbf{mod}24\right)\cdot 16\right)\mathbf{mod}24\\ \text{•}& & \text{Examine term}\\ & & \left({128}^{28}\right)\mathbf{mod}24\\ \text{•}& & \text{Apply the}\text{power}\text{mod rule:}{a}^b\mathbf{mod}m={\left(a\mathbf{mod}m\right)}^b\mathbf{mod}m\\ & & \left({\left(128\mathbf{mod}24\right)}^{28}\right)\mathbf{mod}24\\ \text{•}& & \text{Examine term}\\ & & 128\mathbf{mod}24\\ \text{•}& & \text{Evaluate modulo, which is the remainder of}128\text{divided by}24\\ & & 8\\ \text{•}& & \text{This gives:}\\ & & \left(8^{28}\right)\mathbf{mod}24\\ \text{•}& & \text{Apply exponent rule:}{a}^{nm+c}={\left(a^n\right)}^m\cdot a^c\\ & & \left({\left(8^3\right)}^9\cdot 8\right)\mathbf{mod}24\\ \text{•}& & \text{Evaluate inside exponents}\\ & & \left({512}^9\cdot 8\right)\mathbf{mod}24\\ \text{•}& & \text{Apply the}\text{product}\text{mod rule:}\left(a\cdot b\right)\mathbf{mod}m=\left(\left(a\mathbf{mod}m\right)\cdot \left(b\mathbf{mod}m\right)\right)\mathbf{mod}m\\ & & \left(\left({512}^9\mathbf{mod}24\right)\cdot 8\right)\mathbf{mod}24\\ \text{•}& & \text{Examine term}\\ & & \left({512}^9\right)\mathbf{mod}24\\ \text{•}& & \text{Apply the}\text{power}\text{mod rule:}{a}^b\mathbf{mod}m={\left(a\mathbf{mod}m\right)}^b\mathbf{mod}m\\ & & \left({\left(512\mathbf{mod}24\right)}^9\right)\mathbf{mod}24\\ \text{•}& & \text{Examine term}\\ & & 512\mathbf{mod}24\\ \text{•}& & \text{Evaluate modulo, which is the remainder of}512\text{divided by}24\\ & & 8\\ \text{•}& & \text{This gives:}\\ & & \left(8^9\right)\mathbf{mod}24\\ \text{•}& & \text{Apply exponent rule:}{a}^{nm}={\left(a^n\right)}^m\\ & & \left({\left(8^3\right)}^3\right)\mathbf{mod}24\\ \text{•}& & \text{Evaluate inside exponents}\\ & & \left({512}^3\right)\mathbf{mod}24\\ \text{•}& & \text{Apply the}\text{power}\text{mod rule:}{a}^b\mathbf{mod}m={\left(a\mathbf{mod}m\right)}^b\mathbf{mod}m\\ & & \left({\left(512\mathbf{mod}24\right)}^3\right)\mathbf{mod}24\\ \text{•}& & \text{Examine term}\\ & & 512\mathbf{mod}24\\ \text{•}& & \text{Evaluate modulo, which is the remainder of}512\text{divided by}24\\ & & 8\\ \text{•}& & \text{This gives:}\\ & & \left(8^3\right)\mathbf{mod}24\\ \text{•}& & \text{Evaluate inside exponents}\\ & & \left(512\right)\mathbf{mod}24\\ \text{•}& & \text{Evaluate modulo, which is the remainder of}512\text{divided by}24\\ & & 8\\ \text{•}& & \text{This gives:}\\ & & \left(8\cdot 8\right)\mathbf{mod}24\\ \text{•}& & \text{Simplify}\\ & & 64\mathbf{mod}24\\ \text{•}& & \text{Evaluate modulo, which is the remainder of}64\text{divided by}24\\ & & 16\\ \text{•}& & \text{This gives:}\\ & & \left(16\cdot 16\right)\mathbf{mod}24\\ \text{•}& & \text{Simplify}\\ & & 256\mathbf{mod}24\\ \text{•}& & \text{Evaluate modulo, which is the remainder of}256\text{divided by}24\\ & & 16\\ \text{•}& & \text{This gives:}\\ & & 16+\left(2^{201}\mathbf{mod}24\right)\\ \text{•}& & \text{Examine term}\\ & & \left(2^{201}\right)\mathbf{mod}24\\ \text{•}& & \text{Apply exponent rule:}{a}^{nm+c}={\left(a^n\right)}^m\cdot a^c\\ & & \left({\left(2^7\right)}^{28}\cdot 2^5\right)\mathbf{mod}24\\ \text{•}& & \text{Evaluate inside exponents}\\ & & \left({128}^{28}\cdot 32\right)\mathbf{mod}24\\ \text{•}& & \text{Apply the}\text{product}\text{mod rule:}\left(a\cdot b\right)\mathbf{mod}m=\left(\left(a\mathbf{mod}m\right)\cdot \left(b\mathbf{mod}m\right)\right)\mathbf{mod}m\\ & & \left(\left({128}^{28}\mathbf{mod}24\right)\cdot \left(32\mathbf{mod}24\right)\right)\mathbf{mod}24\\ \text{•}& & \text{Examine term}\\ & & \left({128}^{28}\right)\mathbf{mod}24\\ \text{•}& & \text{Apply the}\text{power}\text{mod rule:}{a}^b\mathbf{mod}m={\left(a\mathbf{mod}m\right)}^b\mathbf{mod}m\\ & & \left({\left(128\mathbf{mod}24\right)}^{28}\right)\mathbf{mod}24\\ \text{•}& & \text{Examine term}\\ & & 128\mathbf{mod}24\\ \text{•}& & \text{Evaluate modulo, which is the remainder of}128\text{divided by}24\\ & & 8\\ \text{•}& & \text{This gives:}\\ & & \left(8^{28}\right)\mathbf{mod}24\\ \text{•}& & \text{Apply exponent rule:}{a}^{nm+c}={\left(a^n\right)}^m\cdot a^c\\ & & \left({\left(8^3\right)}^9\cdot 8\right)\mathbf{mod}24\\ \text{•}& & \text{Evaluate inside exponents}\\ & & \left({512}^9\cdot 8\right)\mathbf{mod}24\\ \text{•}& & \text{Apply the}\text{product}\text{mod rule:}\left(a\cdot b\right)\mathbf{mod}m=\left(\left(a\mathbf{mod}m\right)\cdot \left(b\mathbf{mod}m\right)\right)\mathbf{mod}m\\ & & \left(\left({512}^9\mathbf{mod}24\right)\cdot 8\right)\mathbf{mod}24\\ \text{•}& & \text{Examine term}\\ & & \left({512}^9\right)\mathbf{mod}24\\ \text{•}& & \text{Apply the}\text{power}\text{mod rule:}{a}^b\mathbf{mod}m={\left(a\mathbf{mod}m\right)}^b\mathbf{mod}m\\ & & \left({\left(512\mathbf{mod}24\right)}^9\right)\mathbf{mod}24\\ \text{•}& & \text{Examine term}\\ & & 512\mathbf{mod}24\\ \text{•}& & \text{Evaluate modulo, which is the remainder of}512\text{divided by}24\\ & & 8\\ \text{•}& & \text{This gives:}\\ & & \left(8^9\right)\mathbf{mod}24\\ \text{•}& & \text{Apply exponent rule:}{a}^{nm}={\left(a^n\right)}^m\\ & & \left({\left(8^3\right)}^3\right)\mathbf{mod}24\\ \text{•}& & \text{Evaluate inside exponents}\\ & & \left({512}^3\right)\mathbf{mod}24\\ \text{•}& & \text{Apply the}\text{power}\text{mod rule:}{a}^b\mathbf{mod}m={\left(a\mathbf{mod}m\right)}^b\mathbf{mod}m\\ & & \left({\left(512\mathbf{mod}24\right)}^3\right)\mathbf{mod}24\\ \text{•}& & \text{Examine term}\\ & & 512\mathbf{mod}24\\ \text{•}& & \text{Evaluate modulo, which is the remainder of}512\text{divided by}24\\ & & 8\\ \text{•}& & \text{This gives:}\\ & & \left(8^3\right)\mathbf{mod}24\\ \text{•}& & \text{Evaluate inside exponents}\\ & & \left(512\right)\mathbf{mod}24\\ \text{•}& & \text{Evaluate modulo, which is the remainder of}512\text{divided by}24\\ & & 8\\ \text{•}& & \text{This gives:}\\ & & \left(8\cdot 8\right)\mathbf{mod}24\\ \text{•}& & \text{Simplify}\\ & & 64\mathbf{mod}24\\ \text{•}& & \text{Evaluate modulo, which is the remainder of}64\text{divided by}24\\ & & 16\\ \text{•}& & \text{This gives:}\\ & & \left(16\cdot \left(32\mathbf{mod}24\right)\right)\mathbf{mod}24\\ \text{•}& & \text{Examine term}\\ & & 32\mathbf{mod}24\\ \text{•}& & \text{Evaluate modulo, which is the remainder of}32\text{divided by}24\\ & & 8\\ \text{•}& & \text{This gives:}\\ & & \left(16\cdot 8\right)\mathbf{mod}24\\ \text{•}& & \text{Simplify}\\ & & 128\mathbf{mod}24\\ \text{•}& & \text{Evaluate modulo, which is the remainder of}128\text{divided by}24\\ & & 8\\ \text{•}& & \text{This gives:}\\ & & 16+8\\ \text{•}& & \text{Simplify}\\ & & 24\end{array}$

The Student:-Basics:-ModuloSteps command was introduced in Maple 2024.

For more information on Maple 2024 changes, see Updates in Maple 2024.