What is the least common multiple of 7 and 24?

Apr 7, 2018

$168$

Explanation:

By doing the Prime factorization:
$24 = 2 \times 2 \times 2 \times 3$
$7 = 7$
And since there's isn't any common factor We need all the prime factors listed
$7 \times 2 \times 2 \times 2 \times 3 = 168$

Apr 7, 2018

A teacher will expect the prime number method. Just for the hell of it this is a different approach!

168

Explanation:

We have two numbers ; 24 and 7

I am going to count the 24's. However lets look at this value.

24 can be 'split' into a sum of 7's with a remainder. So each 24 consists of:

$24 = \left(7 + 7 + 7 + 3\right)$

If we sum columns of these we will get the 3 summing to a value into which 7 will divide exactly. When this happens we have found our least common multiple.

REMEMBER WE ARE COUNTING THE 24's

$\text{ count "color(white)("dd") "The 24's}$
$\textcolor{w h i t e}{\text{ddd") 1color(white)("ddd}} \left(\textcolor{w h i t e}{.} 7 + \textcolor{w h i t e}{.} 7 + \textcolor{w h i t e}{.} 7 + \textcolor{w h i t e}{.} 3\right)$
$\textcolor{w h i t e}{\text{ddd") 2color(white)("ddd}} \left(\textcolor{w h i t e}{.} 7 + \textcolor{w h i t e}{.} 7 + \textcolor{w h i t e}{.} 7 + \textcolor{w h i t e}{.} 3\right)$
$\textcolor{w h i t e}{\text{ddd") 3color(white)("ddd}} \left(\textcolor{w h i t e}{.} 7 + \textcolor{w h i t e}{.} 7 + \textcolor{w h i t e}{.} 7 + \textcolor{w h i t e}{.} 3\right)$
$\textcolor{w h i t e}{\text{ddd") 4color(white)("ddd}} \left(\textcolor{w h i t e}{.} 7 + \textcolor{w h i t e}{.} 7 + \textcolor{w h i t e}{.} 7 + \textcolor{w h i t e}{.} 3\right)$
$\textcolor{w h i t e}{\text{ddd") 5color(white)("ddd}} \left(\textcolor{w h i t e}{.} 7 + \textcolor{w h i t e}{.} 7 + \textcolor{w h i t e}{.} 7 + \textcolor{w h i t e}{.} 3\right)$
$\textcolor{w h i t e}{\text{ddd") 6color(white)("ddd}} \left(\textcolor{w h i t e}{.} 7 + \textcolor{w h i t e}{.} 7 + \textcolor{w h i t e}{.} 7 + \textcolor{w h i t e}{.} 3\right)$
color(white)("ddd") 7color(white)("ddd")ul( (color(white)(.)7+color(white)(.)7+color(white)(.)7+color(white)(.)3)larr" Add"
$\textcolor{w h i t e}{\text{ddddddd.}} 49 + 49 + 49 + \underbrace{21}$
$\textcolor{w h i t e}{\text{dddddddddddddddddddd}} \downarrow$
color(white)("dddddddddddddd")" exactly divisible by 7"

We have a count of 7 so the value is $7 \times 24 = 168$