Question

Find$\mathbf{gcd}\mathbf{(}\mathbf{1000}\mathbf{,}\mathbf{625}\mathbf{)}$ and$\mathbf{Icm}\mathbf{(}\mathbf{1000}\mathbf{,}\mathbf{625}\mathbf{)}$ and verify that $\mathbf{gcd}\mathbf{(}\mathbf{1000}\mathbf{,}\mathbf{625}\mathbf{)}\mathbf{·}\mathbf{Icm}\mathbf{(}\mathbf{1000}\mathbf{,}\mathbf{625}\mathbf{)}\mathbf{=}\mathbf{1000}\mathbf{·}\mathbf{625}$.

Short answer

The value of $\gcd \left(1000,625\right)$is 125.

The value of $\mathrm{Icm}\left(1000,625\right)$is $\left(2^3·5^4\right)$.

The verified expression isrole="math" localid="1668517045868" $\gcd \left(1000,625\right)·\mathrm{Icm}\left(1000,625\right)=1000·625$.

Step-by-step solution

Identification of the given data

The given data can be listed below as,

The given expression is, $\gcd \left(1000,625\right)·\mathrm{Icm}\left(1000,625\right)=1000·625$.

Significance of greatest common divisor and least common multiple

Whenever two different positive integers are multiplied with each other, then the result obtained would be equal to the product of its value of the greatest common divisor and least common multiple.

Determination of the greatest common divisor of the pair (1000, 625)

The factors of the given pair of integers can be represented as,

$$\begin{gathered} \begin{array}{rl}(1000,625)=& (125\cdot 8,625)\end{array} \\ =\left(5^3\cdot 2^3,5^4\right) \end{gathered}$$

The greatest common divisor of the pair can be calculated as,

role="math" localid="1668517368718" $$\begin{gathered} \begin{array}{rl}\gcd (1000,625)& =\gcd \left(5^3\cdot 2^3,5^4\right)\end{array} \\ =2^{min(3,0)}\cdot 5^{min(3,4)} \\ =2^0\cdot 5^3 \\ =125 \end{gathered}$$

Thus, the greatest common divisor of the pair is 125.

Determination of the least common multiple of the pair (1000, 625)

The least common multiple of the pair can be calculated as,

$$\begin{gathered} \begin{array}{rl}\mathrm{Icm}(1000,625)& =\mathrm{Icm}\left(5^3\cdot 2^3,5^4\right)\end{array} \\ =2^{max\left(0.3\right)}\cdot 5^{max(3,4)} \\ =\left(2^3\cdot 5^4\right) \end{gathered}$$

Thus, the least common multiple of the pair is $\left(2^3\cdot 5^4\right)$.

Verification of the expression gcd(1000, 625)·Icm(1000, 625)=1000·625

Substitute the values in the left hand side (LHS) of the given expression, we get:

$$\begin{gathered} \begin{array}{rl}\gcd (1000,625)\cdot \mathrm{lcm}(1000,625)& =\left(125\right)\cdot \left(2^3\cdot 5^4\right)\end{array} \\ =\left(125\right)\cdot (8\cdot 625) \\ =(125\cdot 8)\cdot \left(625\right) \\ =(1000\cdot 625) \end{gathered}$$

Thus, the verified expression is $\gcd \left(1000,625\right)·\mathrm{Icm}\left(1000,625\right)=1000·625$.