What does $a\equiv b\pmod n$ mean? What does the $\equiv$ and $b\pmod n$ mean?
for example, what does the following equation mean?
$5x \equiv 7\pmod {24}$? Tomorrow I have a final exam so I really have to know what is it.
 A: 
$a\equiv b\pmod n \Longleftrightarrow$ there is an integer $k$ such that $kn+b=a$

A: $a\equiv b\;(mod\;n) \iff b-a=$ multiple of $n$.
So $5x\equiv 7\;(mod\;24) \iff 5x-7=24k$ for some integer $k$.
A: Let $a=qn+r_{1}$ and $b=pn+r_{2}$, where $0\leq r_{1},r_{2}<n$. Then $$r_{1}=r_{2}.$$ 
$r_{1}$ and $r_{2}$ are remainders when $a$ and $b$ are divided by $n$. 
A: It’s a bit late to be learning a basic definition, but here it is: $a\equiv b\pmod n$ means that $n\mid a-b$, i.e., that $a-b$ is a multiple of $n$. Thus, the congruence $5x\equiv 7\pmod{24}$ means that $24\mid 5x-7$. To solve it, you must find an integer $x$ that makes this true. Since $5\cdot11-7=55-7=48$ is a multiple of $24$, $x\equiv 11\pmod{24}$ is a solution.
A: Another way of thinking: $a\equiv b\pmod n$ means that the remainder of $a$ divided by $n$ is equal to the remainder of $b$ divided by $n$.
A: It means $5x-7$ is divisible by $24$. $(5x-7)/24$ is an integer.
A: As $\lambda(24)=$lcm$(\lambda(3),\lambda(8))=$lcm$(2,2)=2$
Using Carmichael function $a^2\equiv1\pmod{24}$ if $(a,1)=24$
or clearly, $5^2\equiv1\pmod{24}$
So, $5x\equiv7\pmod {24}\implies 5^2x\equiv7\cdot5\pmod{24}\implies x\equiv 35\equiv11\pmod {24} $
