Question

If the product of two integers is 273852711 and their greatestcommon divisor is 23345, what is their least common multiple?

Short answer

The least common multiple$\mathrm{Icm}\left(\mathrm{a},\mathrm{b}\right)$ is $\left(2^4·3^4·5·7^{11}\right)$.

Step-by-step solution

Identification of the given data

The given data can be listed below as,

• The product of two integers is, $\left(a·b\right)=2^7{3}^8{5}^2{7}^{11}$.
• The greatest common divisor is, $gcd\left(a,b\right)=2^3{3}^{4}5$.

Significance of the prime number and greatest common divisor

A specific positive integer greater than 1 and not a prime number refer as a composite number. The largest integer that divides both two integers is called the greatest common divisor of these integers.

Determination of the least common multiple of the two integers

The expression to calculate the least common multiple of the two integers is expressed as follows:

$(a\cdot b)=\gcd (a,b)\cdot \mathrm{Icm}(a,b)$

Substitute the values in the above formula, it gives:

$$\begin{gathered} \left(2^7\cdot 3^8\cdot 5^2\cdot 7^{11}\right)=\left(2^3\cdot 3^4\cdot 5\right)\cdot \mathrm{Icm}(a,b) \\ \mathrm{Icm}(a,b)=\frac{\left(2^7\cdot 3^8\cdot 5^2\cdot 7^{11}\right)}{\left(2^3\cdot 3^4\cdot 5\right)} \\ =\left(2^4\cdot 3^4\cdot 5\cdot 7^{11}\right) \end{gathered}$$

Thus, the least common multiple of the two integers is $\left(2^4\cdot 3^4\cdot 5\cdot 7^{11}\right)$.