Now that we have these three properties of congruences defined, we are better prepared for when we need to solve congruence problems! There are three distinct congruence problem types based on which variable in the expression:
is unknown:
- Suppose only
a
is unknown. Then, from the definition of congruence, we have(a-b) = n*k
, ora = n*k + b
. This means that the answers we seek are all values ofa
for whichn
dividesa-b
, or equivalently all values ofa
for whicha-b
is a multiple ofn
. - Suppose only
b
is unknown. In this case, the procedure for solving forb
is identical to the procedure for solving fora
because we proved the symmetric property of congruences! - Suppose only
n
is unknown. Then, since(a-b) = n*k
by definition of congruence, we are seeking an integerk
in which(a-b)/n
is an integer. In other words, we must find all positive integersn
such thatn
divides(a-b)
. This is equivalent to finding every factor of(a-b)
since they all divide(a-b)
with no remainder.
Thankfully, since we have only three variables present in any given congruence, these three scenarios are the only ones possible (assuming that only one of these three variables is unknown). These next few checkpoints will hopefully boost your confidence in your ability to solve simple congruence problems!
Instructions
Find by hand the smallest number a such that a > 3 and that
is true. Print whether or not the resulting congruence is true.
Instead of solving this by hand, use the symmetric property to come up with an answer for:
Print out whether or not the resulting congruence is true.
Find the largest n for which:
is true, and print whether or not your congruence is true.