Question

How do you find the least common multiple of \[35\] and \[168\]?

Answer 1

Hint: In order to find the solution to the given question that is to find the common multiple of \[35\] and \[168\] Find the prime factorization of \[35\] then Find the prime factorization of \[168\] after this multiply each factor the greater number of times it occurs in prime factors of \[35\] and \[168\] above to find the LCM.

Complete step by step solution:
According to the question, given numbers are as follows:
\[35\] and \[168\]
To find the LCM, follow these steps:
Step 1: Find the prime factors of \[35\]:
To find the prime factorization of the above number start by dividing the number by the first prime number. Here are the first several prime factors: \[2,\text{ }3,\text{ }5,\text{ }7,\text{ }11,\text{ }13,\text{ }17,\text{ }19,\text{ }23,\text{ }29...\]
As we can see \[35\] is not divisible by \[2\] and 3. It’s divisible by 5.
Let's start by dividing \[35\]by \[5\]
\[\Rightarrow 35\div ~5~=7~\]- No remainder! \[5\] is one of the factors!
\[\Rightarrow 7\div ~5~=1.4~\]- There is a remainder. We can't divide by \[5\] evenly anymore.
Let's try the next prime number
\[\Rightarrow 7\div ~7~=1~~~\]- No remainder! \[7~\] is one of the factors!
We can represent this in the following way also:
\[\begin{align}
  & 5\left| \!{\underline {\,
  35 \,}} \right. \\
 & 7\left| \!{\underline {\,
  7 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
The divisor(s) on the right side of the symbol \[\div \] above are the prime factors of the number \[35\]. If we put all of it together, we have the factors \[5\times 7=35\].
Step 2: Find the prime factors of 168:
Let's start by dividing \[168\] by \[2\]
\[\Rightarrow 168\div ~2~=84~\]- No remainder! \[2\] is one of the factors!
\[\Rightarrow 84\div ~2~=42~\]- No remainder! \[2\] is one of the factors!
\[\Rightarrow 42\div ~2~=21~\]- No remainder! \[2\] is one of the factors!
\[\Rightarrow 21\div ~2~=10.5~\]- There is a remainder. We can't divide by \[2\] evenly anymore.
Let's try the next prime number
\[\Rightarrow 21\div ~3~=7~~~\]- No remainder! \[3\] is one of the factors!
\[\Rightarrow 7\div ~3~=2.33\]- There is a remainder. We can't divide by \[3\] evenly anymore.
Let's try the next prime number
\[\Rightarrow 7\div ~7~=1~~~\]- No remainder! \[7~\] is one of the factors!
We can represent this in the following way also:
\[\begin{align}
  & 2\left| \!{\underline {\,
  168 \,}} \right. \\
 & 2\left| \!{\underline {\,
  84 \,}} \right. \\
 & 2\left| \!{\underline {\,
  42 \,}} \right. \\
 & 3\left| \!{\underline {\,
  21 \,}} \right. \\
 & 7\left| \!{\underline {\,
  7 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
The divisor(s) on the right side of the symbol \[\div \] above are the prime factors of the number \[168\]. If we put all of it together, we have the factors \[2\times 2\times 2\times 3\times 7=168\].
Step 3: Multiply each factor the greater number of times it occurs in step 1 and step 2 above to find the LCM.
\[\Rightarrow LCM=2\times 2\times 2\times 3\times 5\times 7\]
\[\Rightarrow LCM=840\]
least common multiple of \[35\] and \[168\] is 840

Note: Students can make mistakes by making division errors, taking a bigger prime number first and taking an even number or the number which is completely wrong and leads to the wrong answer. It’s important to remember that LCM which stands for Least Common Multiple. A multiple is a number you get when you multiply a number by a whole number (greater than 0). A factor is one of the numbers that multiplies by a whole number to get that number.