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Number Theory

# Greatest Common Divisor / Lowest Common Multiple: Level 2 Challenges

Find smallest positive $$a$$ for $$a={ b }^{ 2 }={ c }^{ 3 }={ d }^{ 5 }$$ such that $$a,b,c,d$$ are distinct integers.

\begin{align} (\underbrace{7+7+7+\cdots+7}_{46 \text{ times}}) - (\underbrace{29+29+29+\cdots+29}_{11 \text{ times}}) &= 3 \\\\ (\underbrace{7+7+7+\cdots+7}_{21 \text{ times}}) - (\underbrace{29+29+29+\cdots+29}_{5 \text{ times}}) &= 2 \end{align}

Is it also possible to find positive integers $$m$$ and $$n$$ such that $(\underbrace{7+7+7+\cdots+7}_{m \text{ times}}) - (\underbrace{29+29+29+\cdots+29}_{n \text{ times}}) = 1 ?$

$\large \text{lcm} (2^1 ,2^2 ,2^3,\ldots, 2^n) = 2^{2^{2^2}}$

What is the value of $$n$$ satisfying the equation above? 

Notation: $$\text{lcm}(\cdot)$$ denotes the lowest common multiple function.

For integral choices of $$x$$ and $$y$$, $\text{LCM}(x, y) \leq xy.$

Is the above statement true or false?

Clarification: The $$\text{LCM}$$ is the Lowest Common Multiple of two numbers.

$\large n! + 1 \qquad \qquad (n+1)! + 1$

If $$n$$ is a positive integer, find the greatest common divisor of the two numbers above.

Notation: $$!$$ denotes the factorial notation. For example, $$8! = 1\times2\times3\times\cdots\times8$$.

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