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Property 7

Let a and b be two non-zero integers.

  • If the greatest common divisor of a and b is called the Greatest Common Divisor, then is is denoted by GCD(a, b).
  • The smallest of the positive-common multiples of a and b is called the Least Common Multiple, and it is denoted by LCM(a, b).
  • When determining the GCD and LCM of numbers, the numbers are factorized into their prime factors.
    The smallest powers of the common prime factors are selected for GCD and the largest powers of the common prime factors and the uncommon prime factors are selected for LCM.

Example

\(A=2^3\cdot3^{\enclose{circle}{2}}\cdot5\)
\(B=2^{\enclose{circle}{2}}\cdot3^4\cdot5^2\cdot7\)
\(\operatorname{GCD}(A,B)=2^2\cdot3^2\cdot5\)
\(\operatorname{LCM}(A,B)=2^3\cdot3^4\cdot5^2\cdot7\)
1.
\(A,x,y,z\in\mathbb{Z}^{+}\)
\(A=3x+2=4y+3=5z+4\)
\(\Rightarrow \min(A)=?\)
2.
\(A,a,b,c\in\mathbb{Z}^{+}\)
\(A=3a+1=4b+1=6c+1\)
\(\Rightarrow \min(A)=?\)
3.
\(A,x,y,z\in\mathbb{Z}^{+}\)
\(A=2x-1=3y=5z+2\)
\(\Rightarrow \min(A)=?\)
4.
\(A,x,y,z\in\mathbb{Z}^{+}\)
\(A=2x=4y+6=5z+3\)
\(\Rightarrow \min(A)=?\)
5.
\(A,x,y,z\in\mathbb{Z}^{+}\)
\(A=4x+1=5y-4=6z+7\)
\(\Rightarrow \min(A)=?\)
6.
\(A,x,y,z\in\mathbb{Z}^{+}\)
\(A<550\)
\(A=3x+2=4y+2=5z+2\)
\(\Rightarrow\max(A)=?\)
7.
\(A,a,b\in\mathbb{Z}^{+}\)
\(A>90\)
\(A=3a=5b+4\)
\(\Rightarrow\min(A)=?\)
8.
\(x\in\mathbb{Z}\)
\(\samefrac{200}{x}\in\mathbb{Z}\quad,\quad\samefrac{240}{x}\in\mathbb{Z}\)
\(\Rightarrow \max(x)=?\)
9.
\(x\in\mathbb{Z}\)
\(\samefrac{70}{x}\in\mathbb{Z}\quad,\quad\samefrac{45}{x}\in\mathbb{Z}\)
\(\Rightarrow \max(x)=?\)
10.
\(x\in\mathbb{Z}\)
\(\samefrac{40}{x}\in\mathbb{Z}\quad,\quad\samefrac{48}{x}\in\mathbb{Z}\)
\(\Rightarrow \max(x)=?\)
11.
\(x\in\mathbb{Z}\)
\(\samefrac{-120}{x}\in\mathbb{Z}^+\quad,\quad\samefrac{-150}{x}\in\mathbb{Z}^+\)
\(\Rightarrow \min(x)=?\)
12.
\(x\in\mathbb{Z}\)
\(\samefrac{-60}{x}\in\mathbb{Z}^+\quad,\quad\samefrac{-80}{x}\in\mathbb{Z}^+\)
\(\Rightarrow \min(x)=?\)
13.
\(x\in\mathbb{Z}\)
\(\samefrac{x}{30}\in\mathbb{Z}^+\quad,\quad\samefrac{x}{40}\in\mathbb{Z}^+\)
\(\Rightarrow\min(x)=?\)