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A *modulo form* is a real number which is taken modulo (i.e., within
an integer multiple of) some value `M`. Arithmetic modulo `M`
often arises in number theory. Modulo forms are written
‘`a` `mod` `M`’,
where `a` and `M` are real numbers or HMS forms, and
‘`0 <= a < M`’.
In many applications ‘

To create a modulo form during numeric entry, press the shift-`M`
key to enter the word ‘`mod`’. As a special convenience, pressing
shift-`M` a second time automatically enters the value of ‘`M`’
that was most recently used before. During algebraic entry, either
type ‘`mod`’ by hand or press `M-m` (that’s `META-m`).
Once again, pressing this a second time enters the current modulo.

Modulo forms are not to be confused with the modulo operator ‘`%`’.
The expression ‘`27 % 10`’ means to compute 27 modulo 10 to produce
the result 7. Further computations treat this 7 as just a regular integer.
The expression ‘`27 mod 10`’ produces the result ‘`7 mod 10`’;
further computations with this value are again reduced modulo 10 so that
the result always lies in the desired range.

When two modulo forms with identical ‘`M`’’s are added or multiplied,
the Calculator simply adds or multiplies the values, then reduces modulo
‘`M`’. If one argument is a modulo form and the other a plain number,
the plain number is treated like a compatible modulo form. It is also
possible to raise modulo forms to powers; the result is the value raised
to the power, then reduced modulo ‘`M`’. (When all values involved
are integers, this calculation is done much more efficiently than
actually computing the power and then reducing.)

Two modulo forms ‘`a` `mod` `M`’ and ‘`b` `mod` `M`’
can be divided if ‘`a`’, ‘`b`’, and ‘`M`’ are all
integers. The result is the modulo form which, when multiplied by
‘`b` `mod` `M`’, produces ‘`a` `mod` `M`’. If
there is no solution to this equation (which can happen only when
‘`M`’ is non-prime), or if any of the arguments are non-integers, the
division is left in symbolic form. Other operations, such as square
roots, are not yet supported for modulo forms. (Note that, although
‘`(``a` `mod` `M``)^.5`’ will compute a “modulo square root”
in the sense of reducing
‘`sqrt(a)`’
modulo ‘`M`’, this is not a useful definition from the
number-theoretical point of view.)

It is possible to mix HMS forms and modulo forms. For example, an
HMS form modulo 24 could be used to manipulate clock times; an HMS
form modulo 360 would be suitable for angles. Making the modulo ‘`M`’
also be an HMS form eliminates troubles that would arise if the angular
mode were inadvertently set to Radians, in which case
‘`2@ 0' 0" mod 24`’ would be interpreted as two degrees modulo
24 radians!

Modulo forms cannot have variables or formulas for components. If you
enter the formula ‘`(x + 2) mod 5`’, Calc propagates the modulus
to each of the coefficients: ‘`(1 mod 5) x + (2 mod 5)`’.

You can use `v p` and `%` to modify modulo forms.
See Packing and Unpacking. See Basic Arithmetic.

The algebraic function ‘`makemod(a, m)`’ builds the modulo form
‘`a mod m`’.

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