The Trick of a Mathematical Game. The One’s Digit of the Sum of Two Numbers.
Decipher the trick of the following mathematical magic.
The Rule of the Game
Here is the game.
Pick six natural numbers ($1, 2, 3, \dots$) and place them in the yellow discs of the picture below.
For example, let's say I have chosen the numbers $7, 5, 3, 2, […]

Rank of the Product of Matrices $AB$ is Less than or Equal to the Rank of $A$
Let $A$ be an $m \times n$ matrix and $B$ be an $n \times l$ matrix. Then prove the followings.
(a) $\rk(AB) \leq \rk(A)$.
(b) If the matrix $B$ is nonsingular, then $\rk(AB)=\rk(A)$.
Hint.
The rank of an $m \times n$ matrix $M$ is the dimension of the range […]

$(x^3-y^2)$ is a Prime Ideal in the Ring $R[x, y]$, $R$ is an Integral Domain.
Let $R$ be an integral domain. Then prove that the ideal $(x^3-y^2)$ is a prime ideal in the ring $R[x, y]$.
Proof.
Consider the ring $R[t]$, where $t$ is a variable. Since $R$ is an integral domain, so is $R[t]$.
Define the function $\Psi:R[x,y] \to R[t]$ sending […]

Use Cramer’s Rule to Solve a $2\times 2$ System of Linear Equations
Use Cramer's rule to solve the system of linear equations
\begin{align*}
3x_1-2x_2&=5\\
7x_1+4x_2&=-1.
\end{align*}
Solution.
Let
\[A=[A_1, A_2]=\begin{bmatrix}
3 & -2\\
7& 4
\end{bmatrix},\]
be the coefficient matrix of the system, where $A_1, A_2$ […]

The Product of Distinct Sylow $p$-Subgroups Can Never be a Subgroup
Let $G$ a finite group and let $H$ and $K$ be two distinct Sylow $p$-group, where $p$ is a prime number dividing the order $|G|$ of $G$.
Prove that the product $HK$ can never be a subgroup of the group $G$.
Hint.
Use the following fact.
If $H$ and $K$ […]

Solving a System of Differential Equation by Finding Eigenvalues and Eigenvectors
Consider the system of differential equations
\begin{align*}
\frac{\mathrm{d} x_1(t)}{\mathrm{d}t} & = 2 x_1(t) -x_2(t) -x_3(t)\\
\frac{\mathrm{d}x_2(t)}{\mathrm{d}t} & = -x_1(t)+2x_2(t) -x_3(t)\\
\frac{\mathrm{d}x_3(t)}{\mathrm{d}t} & = -x_1(t) -x_2(t) […]

The Quotient by the Kernel Induces an Injective Homomorphism
Let $G$ and $G'$ be a group and let $\phi:G \to G'$ be a group homomorphism.
Show that $\phi$ induces an injective homomorphism from $G/\ker{\phi} \to G'$.
Outline.
Define $\tilde{\phi}([g])=\phi(g)$ and show that this is well-defined.
Show […]