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Number Theory

Greatest Common Divisor / Lowest Common Multiple

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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