L2 Algebra
Question Bank
Prove that the sum of any two consecutive odd numbers is always a multiple of 4.
The calendar grid shows days in a week (7 columns). Prove that the sum of the numbers in any 2x2 square of dates is always a multiple of 4.
A 3x3 square is placed over the number grid as shown. Prove that the sum of the nine values inside the square is always 9 times the value of the middle square.
Prove that for any 2x2 square on a calendar, the average of the four numbers is always an integer.
Timmy starts filling in a triangle with 3 rows with consecutive multiples of 5 as shown. He notices the sum of the numbers is a multiple of 15. Will this always be the case? Use algebra to prove your result.
Prove that a cuboid with dimensions 2 x 3 x n, as shown in the diagram, always has a surface area that is 2 more than a multiple of 10.
Given that $a, b,$ and $c$ are positive real numbers, prove that if the terms $\log(a)$, $\log(b)$, and $\log(c)$ have a constant difference, i.e. $\log(b) - \log(a) = \log(c) - \log(b),$ then the terms $a, b,$ and $c$ must have a constant ratio, i.e. $\frac{b}{a} = \frac{c}{b}$.
Prove that for any two-digit number, when you reverse the digits and add the new number to the original number, the result will always be a multiple of 11.
Prove that the difference between the squares of any two consecutive odd integers is always a multiple of 8.
For a quadratic equation $ax^2 + bx + c = 0$ with two real roots, prove that the sum of the roots is equal to $-\frac{b}{a}$.
Consider the two quadratic equations:
$x^2+px+q=0$ $x^2+qx+p=0$Where $p$ and $q$ are constants, and $p \ne q$. Show that if these two equations share a common root, then this root must be 1.
Given that $a$ and $b$ are positive numbers not equal to 1, and that $x$ and $y$ are non-zero constants, prove that if $a^x = b^y = (ab)^{xy}$, then it must be true that $x+y=1$.