Question

Find$\mathbf{gcd}\mathbf{(}\mathbf{92928}\mathbf{,}\mathbf{123552}\mathbf{)}$ and$\mathbf{Icm}\mathbf{(}\mathbf{92928}\mathbf{,}\mathbf{123552}\mathbf{)}$ and verify that $\mathbf{gcd}\mathbf{(}\mathbf{92928}\mathbf{,}\mathbf{123552}\mathbf{)}\mathbf{\cdot }\mathbf{Icm}\mathbf{(}\mathbf{92928}\mathbf{,}\mathbf{123552}\mathbf{)}\mathbf{=}\mathbf{92928}\mathbf{\cdot }\mathbf{123552}$. [Hint: First, find the prime factorizations of 92928 and 123552.]

Short answer

The value of$\gcd (92928,123552)$is 1056.

The value of $\mathrm{Icm}(92928,123552)$is.

The verified expression is $\gcd (92928,123552)\cdot \mathrm{Icm}(92928,123552)=92928\cdot 123552$.

Step-by-step solution

Identification of the given data

The given data can be listed below as,

The given expression is,$\gcd (92928,123552)\cdot \mathrm{Icm}(92928,123552)=92928\cdot 123552$ .

Significance of the prime factorization

The term 'prime factorization' of the greatest common divisor (GCD) contains all the common primes in the prime factorizations of two distinct integers, whereas the prime factorization of the least common multiple contains all the primes in the prime factorization of the distinct integers.

Determination of the prime factorization of each integer of the pair (92928,123552)

The prime factorization of each integer of the pair of the integerscan be represented as,

$$\begin{gathered} (92928,123552)=(256\cdot 363,32\cdot 3862) \\ =(256\cdot 3\cdot 121,32\cdot 27\cdot 143) \\ =\left(2^8\cdot 3\cdot {11}^2,2^5\cdot 3^3\cdot 11\cdot 13\right) \end{gathered}$$

From the above calculation, the prime factorization of the pair of integers is $\left(2^8\cdot 3\cdot {11}^2,2^5\cdot 3^3\cdot 11\cdot 13\right)$.

Determination of the greatest common divisor of the pair (92928,123552)

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

$$\begin{gathered} \gcd (92928,123552)=\gcd \left(2^8\cdot 3\cdot {11}^2,2^5\cdot 3^3\cdot 11\cdot 13\right) \\ =2^{min(8,5)}\cdot 3^{min(1,3)}\cdot {11}^{min(2,1)}\cdot {13}^{min(0,1)} \\ =2^5\cdot 3^1\cdot {11}^1\cdot {13}^0 \\ =1056 \end{gathered}$$

Thus, the greatest common divisor of the pair is 1056.

Determination of the least common multiple of the pair (92928,123552)

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

$$\begin{gathered} \mathrm{Icm}(92928,123552)=\mathrm{lcm}\left(2^8\cdot 3\cdot {11}^2,2^5\cdot 3^3\cdot 11\cdot 13\right) \\ =2^{max(8,5)}\cdot 3^{max(1,3)}\cdot {11}^{max(2,1)}\cdot {13}^{max(0,1)} \\ =2^8\cdot 3^3\cdot {11}^2\cdot {13}^1 \\ =10872576 \end{gathered}$$

Thus, the least common multiple of the pair is $10872576$.

Verification of the expression gcd(92928,123552)⋅Icm(92928,123552)=92928⋅123552

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

$$\begin{gathered} \gcd (92928,123552)\cdot \mathrm{lcm}(92928,123552)=\left(2^5\cdot 3\cdot 11\right)\cdot \left(2^8\cdot 3^3\cdot {11}^2\cdot 13\right) \\ =\left(2^5\cdot 3\cdot {11}^2\cdot 2^3\right)\cdot \left(2^5\cdot 3^3\cdot 11\cdot 13\right) \\ =(92928\cdot 123552) \end{gathered}$$

Thus, the verified expression is $\gcd (92928,123552)\cdot \mathrm{Icm}(92928,123552)=92928\cdot 123552$.