Key Fact
7446

The product of a rational number and an irrational number is irrational.
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Key Fact
7695

For a fraction to be in lowest terms (or, to be written in its simplest form), its numerator and denominator must be relatively prime.
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Key Fact
ABAD

The square of any even integer is of form $4k$.
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Key Fact
BAB3

The square of any odd integer is of form $4k+1$.
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Key Fact
988C

The product of four consecutive natural numbers is never a perfect square.
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Key Fact
9A34

$$1001=7\cdot11\cdot13$$
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Key Fact
B664

$$n\cdot n!=(n+1)!-n!$$
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Key Fact
969B

$$\frac{1}{n\cdot(n+1)}=\frac{1}{n}-\frac{1}{n+1}$$
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Math

Key Fact
CE81

$$\frac{1}{n \cdot (n+m)}=\frac{1}{m} \left( \frac{1}{n}-\frac{1}{n+m} \right)$$
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Math

Key Fact
B945

$$\frac{1}{n(n+1)(n+2)}=\frac{1}{2}\cdot \left[ \frac{1}{n(n+1)} - \frac{1}{(n+1)(n+2)} \right]$$
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Math

Key Fact
BCC8

$$\frac{n}{(n+1)!}=\frac{1}{n!}-\frac{1}{(n+1)!}$$
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Math

Key Fact
965A

$$n^4+n^2+1=(n^2+1-n)\cdot(n^2+1+n)$$
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Key Fact
E856

$$(n+1)^2-(n+1)+1=n^2+n+1$$
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Key Fact
7C45

$\small{\textbf{Sophie-Germain Identity}}$: $$\begin{align}a^4+4b^4&\cssId{Step1}{=\left[(a+b)^2+b^2\right]\left[(a-b)^2+b^2\right]}\\&\cssId{Step1}{=(a^2+2ab+2b^2)(a^2-2ab+2b^2)} \end{align}$$
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Math

Key Fact
EA1C

$$x^4+y^4+z^4=(x^2+y^2+z^2)^2-2(x^2y^2+y^2z^2+x^2z^2)$$
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Math

Key Fact
3E17

For odd $n \in N$, $$a^n+b^n=(a+b)(a^{n-1}-a^{n-2} \cdot b \,+\,...\,-\,a \cdot b^{n-2}+b^{n-1})$$
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Math

Key Fact
2545

For all $n \in N$, $$a^n-b^n=(a-b)(a^{n-1}+a^{n-2} \cdot b \,+\,...\,+\,a \cdot b^{n-2}+b^{n-1})$$
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Math

Key Fact
A376

For $a=k\cdot (2k+1)$ where $k \in N$,$$a^2+(a+1)^2+⋯+(a+k)^2\\=(a+k+1)^2+(a+k+2)^2+⋯+(a+2k)^2$$
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Key Fact
3278

A composite number is a positive integer greater than 1 that has more than two positive divisors.
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Key Fact
2DCC

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
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Key Fact
87D4

All positive integers greater than 1 are either prime or composite.
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Key Fact
9E11

1 is the only positive integer that is neither prime nor composite.
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Key Fact
A478

The only even prime number is 2.
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Key Fact
2AB2

No prime number greater than 5 ends in a 5.
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Key Fact
DAEA

List of prime numbers up to 100: $$2,\,3,\,5,\,7,\,11,\,13,\,17,\,19,\,23,\,29,\,31,\,37,\,41,$$$$\,43,\,47,\,53,\,59,\,61,\,67,\,71,\,73,\,79,\,83,\,89,\,97$$
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Math

Key Fact
9ECE

Every integer greater than 1 has a unique prime factorization up to the order of the factors.
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Math

Key Fact
525C

If $n$ is a composite number, then it must be divisible by a prime $p$ such that $p \le \sqrt{n}$.
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Math

Key Fact
163E

$\small{\textbf{ Wilson's Theorem}}:\\[5pt]\text{ A positive integer }p>1 \text{ is prime if and only if }$ $$(p-1)! \equiv -1 \hspace{0.3cm} (\text{mod } p)$$
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Math

Key Fact
5221

$\small{\textbf{ Bertrand's Postulate}}:\\[5pt]\text{ For any integer } n>3 \text{, there always exists at least one prime number} \\ \, p \text{ with } $ $$n < p < 2n-2$$
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Math

Key Fact
24D6

$\small{\textbf{ Bonse's Inequality}}:\\[5pt]\text{ If } p_1, \, ...,\, p_n, \,p_{n+1} \text{ are the smallest } n+1 \text{ prime numbers and } n \ge4 \text{,} \\ \, \text{then}$ $$p_1 \, \cdot \, ... \, \cdot \, p_n < \, p_{n+1}^2 $$
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Math

Key Fact
C181

If $p$ is a prime number, then the prime power $p^a$ has $a+1$ divisors.
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Math

Key Fact
8A33

The square of any odd integer leaves remainder 1 upon division by 8.
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Math

Key Fact
7B55

When $n$ is a positive even integer,$$n⋅(n+4)⋅(n+8)⋅(n+12)$$is divisible by 8.
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Math

Key Fact
8C41

$n^3-n$ is always divisible by 6, where $n \in Z$.
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Key Fact
8632

The product of any $n$ consecutive integer is always divisible by $n!$
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Math

Key Fact
EADA

If $n$ is a positive integer,$$(n+1)\cdot(n+2)⋅…⋅(2n)$$is divisible by $2^n$.
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Key Fact
A4A7

The binomial expansion of $(x+y)^n$ has $n+1$ terms.
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Key Fact
3845

The binomial expansion of $$(a_1+a_2+⋯+a_{r-1}+a_r )^n$$has $\large {n+r-1 \choose r-1}$ terms.
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Key Fact
D8CD

$$1+2+3\,+\,...\,+\,n=\frac{n(n+1)}{2}$$
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Key Fact
DEA5

$$1^2+2^2+3^2\,+\,...\,+\,n^2=\frac{n(n+1)(2n+1)}{6}$$
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Math

Key Fact
7B38

$$1^3+2^3+3^3+...+\,n^3=\left[\frac{n(n+1)}{2}\right]^2$$
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Math

Key Fact
2EBB

$$1^4+2^4+3^4+...+\,n^4=\frac{pst}{30}$$ where $p=n(n+1)$, $s=2n+1$, and $t=3p-1$.
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Math

Key Fact
89DD

$$\sum_{k=1}^n k \cdot (k+1) \cdot (k+2) = \frac{n\cdot(n+1)\cdot(n+2)\cdot(n+3)}{4}$$
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Math

Key Fact
ECD9

There are $$\frac{n\cdot(n+1)\cdot(2n+1)}{6}$$squares on an $n\times n$ chessboard.
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Math

Key Fact
A86B

The $\small{\textbf{Arithmetic Mean}}$ ($\small{\textbf{AM}}$) of positive real numbers $x_1, x_2, ...,x_n$ is defined as$$\frac{x_1 + x_2 + ... + x_n}{n}$$
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Math

Key Fact
B65C

The $\small{\textbf{Geometric Mean}}$ ($\small{\textbf{GM}}$) of positive real numbers $x_1, x_2, ...,x_n$ is defined as$$\root n \of {x_1 \cdot x_2 \cdot ...\cdot x_n}$$
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Math

Key Fact
D477

For any set of positive real numbers $x_1,\ldots,x_n$, the arithmetic mean is greater than or equal to the geometric mean. That is,$$\frac{x_1 + x_2 + ... + x_n}{n}\ge\root n \of {x_1 \cdot x_2 \cdot ...\cdot x_n}$$
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Math

Key Fact
DDC5

The $\small{\textbf{Harmonic Mean}}$ ($\small{\textbf{HM}}$) of positive real numbers $x_1, x_2, ...,x_n$ is defined to be$$\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}}$$
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Math

Key Fact
29D9

For any set of positive real numbers $x_1,\ldots,x_n$, the arithmetic mean is greater than or equal to the harmonic mean. That is,$$\frac{x_1 + x_2 + ... + x_n}{n}\ge\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}}$$
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Math

Key Fact
5375

For non-zero real numbers $a$ and $b$,$$\frac{a}{b}+\frac{b}{a}\ge2$$
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Math

Key Fact
1E7C

For real numbers $a$ and $b$, $$a^2+b^2+1 \ge ab+a+b$$
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Key Fact
3981

For real numbers $a$ and $b$, $$a^4+b^4+8 \ge 8ab$$
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Math

Key Fact
1EB9

For positive real numbers $a$, $b$, and $c$, $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\le\frac{a^2+b^2+c^2}{a\cdot b\cdot c}$$
Ah
Math

Key Fact
D7B8

For positive real numbers $a$, $b$, and $c$, $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\ge3$$.
Ah
Math

Key Fact
2483

For real numbers $a,b,c \ge 0$, $$(a+b)\cdot(a+c)\cdot(b+c) \ge 8abc$$
Ah
Math

Key Fact
6C13

For real numbers $a,b,c \ge 0$, $$(a^2+1)\cdot(b^2+1)\cdot(c^2+1) \ge 8abc$$
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Math

Key Fact
7185

For real numbers $a,b,c\ge0$, $$ab+ac+bc\ge a \cdot \sqrt{bc}+b \cdot \sqrt{ac}+c \cdot \sqrt{ab}$$
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Math

Key Fact
8C62

For positive real numbers $a$, $b$, $c$, and $d$, $$(a+b+c+d)\cdot \left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d} \right)\ge16$$
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Math

Key Fact
913D

$$(1+x)^n\ge(1+n\cdot x)$$
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Math

Key Fact
ADBA

$\small{\textbf{Cauchy–Schwarz Inequality}}:\\[0pt]$ $\text{For all sequences of real numbers $a_i$ and $b_i$, we have}$ $$\left(\sum_{i=1}^n a_i^2\right)\cdot\left(\sum_{i=1}^n b_i^2\right)\ge\left(\sum_{i=1}^n a_i\cdot b_i\right)^2$$
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Math

Key Fact
2169

$\small{\textbf{ Chebyshev Inequality}}:\\[5pt]\text{ Let }x_1, x_2,..., x_n \text{ and } y_1, y_2,..., y_n \text{ be two sequences of real }\\ \text{ numbers, such that } x_1 \le x_2 \le \cdots \le x_n \text{ and } y_1 \le y_2 \le \cdots \le y_n. \\ \text{ Then, }$$$\frac{(x_1+x_2+...+x_n)(y_1+y_2+...+y_n)}{n} \le x_1y_1+x_2y_2+...+x_ny_n$$
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Math

Key Fact
E1B9

Let $y=k$ be any line which intersects $y=ax^2+bx+c$ at two points, say $P$ and $Q$, where $a, b, c, k \in R$ and $a \neq 0$. Then the abscissa of the midpoint of the line segment $PQ$ is the abscissa of the vertex of the parabola.
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Math

Key Fact
72A4

$\small{\textbf{De Moivre's Formula}}:$ For any angle $\alpha$ and for any integer $n$, $$(\text{cos } \alpha + i\cdot \text{sin } \alpha)^n=\text{cos } n\alpha + i\cdot \text{sin } n\alpha$$
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Math

Key Fact
5DB3

$\small{\textbf{Fibonacci Numbers}}:\\[0pt]\text{Sequence defined recursively by } F_1=F_2=1 \text{ and } F_{n+2}=F_{n+1}+F_n \\ \text{for all } n \in N.$
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Math

Key Fact
47EB

A finite series is given by all the terms of a finite sequence, added together.
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Math

Key Fact
13D3

An infinite series is given by all the terms of an infinite sequence, added together.
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Math

Key Fact
EAA4

The $n$th partial sum of a series is the sum of the first $n$ terms.
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Math

Key Fact
3332

The sequence of partial sums of a series sometimes tends to a real limit. If this happens, we say that this limit is the sum of the series.
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Math

Key Fact
2B69

The sequence of partial sums of a series sometimes tends to a real limit. If this does not happen, we say that the series has no sum.
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Math

Key Fact
C62B

A series can have a sum only if the individual terms tend to zero. But there are some series (with individual terms tending to zero) that do not have sums.
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Math

Key Fact
1529

The sum of an infinite series of the form $$a_1+a_1\cdot r+a_1\cdot r^2+a_1\cdot r^3+...$$ is $\large{\frac{a_1}{1-r}}$ where $\left| r \right| < 1$.
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Math

Key Fact
E6E7

Relative error = error / measurement
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Math

Key Fact
8B56

The bisector of an angle of the triangle divides the opposite side into segments proportional to the other two sides.
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Math

Key Fact
B4C3

The sum of the exterior angles of any (convex) polygon is a constant, namely two straight angles (360$^{\circ}$).
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Math

Key Fact
C4CC

Of all triangles inscribable in a semi-circle, with the diameter as base, the one with the greatest area is the one with the largest altitude (the radius); that is, isosceles triangle.
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Math

Key Fact
8146

Two equal circles in a plane cannot have only one common tangent.
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Math

Key Fact
B13A

The number of lattice points on the segment from the origin to $(a,b)$ is one more than the greatest common factor of $a$ and $b$.
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Math

Key Fact
4BC8

A triangle is a right triangle if and only if one of its medians is half as long as the side to which it is drawn.
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Math