Question

How do you find the least common multiple of these two expressions $ 10{{y}^{2}}{{w}^{6}}{{u}^{5}},6{{y}^{4}}{{w}^{7}} $ ? \[\]

Answer 1

Hint: We recall the least common multiple of two numbers and then two expressions. We first find the factors in both expressions. We first extract unique factors in both the expression and then we extract the common factors with the highest power on them in comparison in both expressions. We multiply the extracted factors to get the least common multiple of the given two expressions. \[\]


Complete step by step answer:
We know that the least common multiple of two numbers abbreviated as lcm is the smallest multiple that is common to both the numbers. Similarly, the lcm of two algebraic expressions is an algebraic expression which is the smallest multiple and common to both the expression. We are asked in the question to find the lcm of $ 10{{y}^{2}}{{w}^{6}}{{u}^{5}},6{{y}^{4}}{{w}^{7}} $ . We can write the given expression as

\[\begin{align}
  & 10{{y}^{2}}{{w}^{6}}{{u}^{5}}=2\cdot 5\cdot {{y}^{2}}\cdot {{w}^{6}}\cdot {{u}^{5}} \\
 & 6{{y}^{4}}{{w}^{7}}=2\cdot 3\cdot {{y}^{4}}\cdot {{w}^{7}} \\
\end{align}\]

We see the factors of $ 10{{y}^{2}}{{w}^{6}}{{u}^{5}} $ are $ 2,5,y,w,u,{{y}^{2}},{{w}^{2}},{{w}^{3}},{{w}^{4}},{{w}^{5}},{{w}^{6}},{{u}^{2}},{{u}^{3}},{{u}^{4}},{{u}^{5}} $ and the factors of $ 6{{y}^{4}}{{w}^{7}} $ are \[2,3,y,w,{{y}^{2}},{{y}^{3}},{{y}^{4}},{{w}^{2}},{{w}^{3}},{{w}^{4}},{{w}^{5}},{{w}^{6}},{{w}^{7}}\]. Any multiple of both $ 10{{y}^{2}}{{w}^{6}}{{u}^{5}},6{{y}^{4}}{{w}^{7}} $ have to be divided by all the factors of $ 10{{y}^{2}}{{w}^{6}}{{u}^{5}},6{{y}^{4}}{{w}^{7}} $ . If the multiple is divided by the term with highest power, it automatically divided by the lower powers of that term. So we need only the term with highest power to find lcm.\[\]
So we first extract the unique factor of $ 10{{y}^{2}}{{w}^{6}}{{u}^{5}} $ that is 5 and $ {{u}^{5}} $ since $ {{u}^{5}} $ is the highest power and also is a unique factor of $ 10{{y}^{2}}{{w}^{6}}{{u}^{5}} $ . We extract the unique factor of $ 6{{y}^{4}}{{w}^{7}} $ that is 3. \[\]
We extract the factor 2 which is common to both once since its power on both the expression is equal. The factor $ y $ is common factor to both the expressions and so we extract the term with highest power of $ y $ that is $ {{y}^{4}} $ from $ 6{{y}^{4}}{{w}^{7}} $ . Similarly since $ w $ is a common factor both the expressions and so we extract the term with highest power of $ w $ that is $ {{w}^{7}} $ from $ 6{{y}^{4}}{{w}^{7}} $ . The least common multiple of both the expressions is the product of extracted factor that is
\[2\times 3\times 5\times {{u}^{5}}\times {{y}^{4}}\times {{w}^{7}}=30{{y}^{4}}{{u}^{5}}{{w}^{7}}\].

Note:
We note that if $ p\left( x \right),q\left( x \right) $ are algebraic expressions that do not share any common factors two then their least common multiple is $ p\left( x \right)\cdot q\left( x \right) $ . We can find the greatest common divisor of two expressions by taking the product of common factors such that each common factor has lowest power on them in comparison in the two expressions which here is $ 2\times {{y}^{2}}\times {{w}^{6}}=2{{y}^{2}}{{w}^{6}} $ .