# Use the prime factorizations a = 2^4 xx 3^4 xx 5^2 xx 7^3 and b = 2^2 xx 3 xx 5^3 xx 11 to find the prime factorization of the following. (a) LCM(a, b) (b) GCF(a, b)

Question
Polynomial factorization
Use the prime factorizations $$\displaystyle{a}={2}^{{4}}\times{3}^{{4}}\times{5}^{{2}}\times{7}^{{3}}{\quad\text{and}\quad}{b}={2}^{{2}}\times{3}\times{5}^{{3}}\times{11}$$ to find the prime factorization of the following.
(a) LCM(a, b)
(b) GCF(a, b)

2021-02-01
Step 1
Given
$$\displaystyle{a}={2}^{{4}}\times{3}^{{4}}\times{5}^{{2}}\times{7}^{{3}}$$
$$\displaystyle{b}={2}^{{2}}\times{3}\times{5}^{{3}}\times{11}$$
Step 2
LCM(a,b)= Product of the greatest power of each prime factor involved in the numbers
$$\displaystyle{L}{C}{M}{\left({a},{b}\right)}={2}^{{4}}\times{3}^{{4}}\times{5}^{{3}}\times{7}^{{3}}\times{11}$$
GCF(a,b)= Product of the smallest power of each common prime factor in the numbers
$$\displaystyle{G}{C}{F}{\left({a},{b}\right)}={2}^{{2}}\times{3}\times{5}^{{2}}$$

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