UPSKILL MATH PLUS

Learn Mathematics through our AI based learning portal with the support of our Academic Experts!

Two integers, $$a$$ and $$b$$, are said to be congruent modulo $$n$$ if and only if their difference is divisible by $$n$$. It can be written as

$$a \equiv b (mod \ n)$$

In other words, the two integers, $$a$$ and $$b$$, is the difference of the integer multiple of congruence modulo $$n$$. That is, $$b - a = kn$$ for some integer $$k$$.

Here, $$n$$ is called the modulus of the congruence.
Example:
$$39 \equiv 4 (mod \ 5)$$. Here, $$5$$ is the modulus of the congruence because $$39 - 4 = 35$$ is divisible by $$5$$.
Consider the following examples on how to find the congruence modulo.
Example:
1. Find $$3 (mod \ 2)$$

Solution:

Consider the possible remainders for the number with mod $$2$$ are $$0$$, $$1$$.

To find the remainder of the given number, start at $$0$$ and go through $$3$$ numbers in a clockwise direction. The $$3$$ numbers would go in a cycle starting from $$0$$. They are $$1$$, $$0$$, $$1$$. Since the cycle ends at the remainder of $$1$$, the answer for $$3 (mod \ 2)$$ is $$1$$.

Therefore, $$3 \equiv 1 (mod \ 2)$$

2. Find $$-3 (mod \ 2)$$

Solution:

Consider the possible remainders for the number with mod $$2$$ are $$0$$, $$1$$.

To find the remainder of the given number, start at $$0$$ and go through $$3$$ numbers in a anticlockwise direction(because the number has a negative sign). The $$3$$ numbers would go in a cycle starting from $$0$$. They are $$1$$, $$0$$, $$1$$. Since the cycle ends at the remainder of $$1$$, the answer for $$-3 (mod \ 2)$$ is $$1$$.

Therefore, $$-3 \equiv 1 (mod \ 2)$$