Number Theory
# Greatest Common Divisor / Lowest Common Multiple

If \(x\) and \(y\) are integers, what is the smallest possible positive value of \(30x+18y?\)

\[\huge \color{red}{2}, \color{green}{-4}\]

Find the lowest common multiple of the two numbers above.

\[\LARGE \color{blue}{0.\bar{3}}, \color{red}{0.\bar{5}}\]

What is the lowest common multiple of the two numbers given above?

Note that \(0. \overline{ABC} = 0.ABCABCABC\ldots \).

How many positive integers \(a\) are there such that 2027 divided by \(a\) leaves a remainder of 7?

Evaluate \( \gcd ( 19 ! + 19, 20! + 19 ) \).

**Details and assumptions**

The number \( n!\), read as **n factorial**, is equal to the product of all positive integers less than or equal to \(n\). For example, \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\).

How many ordered pairs of positive integers \( (m, n)\) satisfy

\[ \gcd (m^3, n^2) = 2^2 \cdot 3^2 \text{ and } \text{lcm}(m^2, n^3) = 2^4 \cdot 3^4 \cdot 5^6? \]

This problem is shared by Muhammad A.

**Details and assumptions**

\( \gcd(a, b) \) and \( \text{lcm}(a, b) \) denote the greatest common divisor and least common multiple of \( a \) and \( b \), respectively.