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Greatest Common Divisor / Lowest Common Multiple
What is the largest number that can divide two numbers without a remainder? What is the smallest number that is divisible by two numbers without a remainder?
Find smallest positive \(a\) for \(a={ b }^{ 2 }={ c }^{ 3 }={ d }^{ 5 }\) such that \(a,b,c,d\) are distinct integers.
by
shivamani patil
\[\begin{align} (\underbrace{7+7+7+\cdots+7}_{46 \text{ times}}) - (\underbrace{29+29+29+\cdots+29}_{11 \text{ times}}) &= 3 \\\\ (\underbrace{7+7+7+\cdots+7}_{21 \text{ times}}) - (\underbrace{29+29+29+\cdots+29}_{5 \text{ times}}) &= 2 \end{align}\]
Is it also possible to find positive integers \(m\) and \(n\) such that \[ (\underbrace{7+7+7+\cdots+7}_{m \text{ times}}) - (\underbrace{29+29+29+\cdots+29}_{n \text{ times}}) = 1 ? \]
by
Pi Han Goh
\[ \large \text{lcm} (2^1 ,2^2 ,2^3,\ldots, 2^n) = 2^{2^{2^2}} \]
What is the value of \(n\) satisfying the equation above? \[\]
Notation: \(\text{lcm}(\cdot) \) denotes the lowest common multiple function.
by
Pi Han Goh
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.
by
Zandra Vinegar
\[ \large n! + 1 \qquad \qquad (n+1)! + 1 \]
If \(n\) is a positive integer, find the greatest common divisor of the two numbers above.
\[\]Notation: \(!\) denotes the factorial notation. For example, \(8! = 1\times2\times3\times\cdots\times8 \).
by
Anuj Shikarkhane