Question

# For each of the following pairs, write a and b in standard form and use these factorizations to find (a, b) and [a, b]. a = 3780, b = 16,200

Polynomial factorization
For each of the following pairs, write a and b in standard form and use these factorizations to find (a, b) and [a, b].
a = 3780, b = 16,200

2021-05-08
Step 1
Given :
a = 3780 and b = 16200.
To write a and b in standard form and to find (a, b) and [a,b].
Step 2
Standard form of n :
Suppose $$p_{1},p_{2},...,p_{r}$$ are the distinct prime factors of n arranged in order $$p_{1} then < ... < \(p_{r}$$
$$n=p_{1}^{m_{1}}p_{2}^{m_{2}}...p_{r}^{m_{r}}$$ where each $$m_{i}$$ is a positive integer.
(a, b) can be found by finding the product of all the common prime factors, with each common factor raised to the least power to which it appears in either factorization.
[a, b] can be found by finding the product of all distinct prime factors that appears in the standard form of either a or b, with each factor raised to the greatest power to which it appears in either factorization.
Step 3
Now,
$$a=3780=2^{2}\times 3^{3}\times 5^{1}\times 7^{1}$$
$$b=16200 = 2^{3}\times 3^{4}\times 5^{2}$$
Hence, the standard form of a and b are
$$a=3780=2^{2}\times 3^{3}\times 5\times 7$$
$$b=16200 = 2^{3}\times 3^{4}\times 5^{2}$$
Use these factorization to find (a, b) and [a,b]
$$(a,b)=(3780, 16200)=2^{2}\times 3^{3}\times 5=540$$.
$$[a,b]=[3780, 16200]=2^{3}\times 3^{4}\times 5^{2}\times 7=113400$$.
Hence, (a, b) = 540 and [a, b] = 113400.