Question

What is the least common multiple of \[32,15\& 36\]?

Answer 1

Hint: LCM (Least common multiple) $ = $ Product of the greatest power of each prime factor, involved in the numbers or product of each prime factors which are common in all three number and common in only two numbers and present only in one given number.

Complete step-by-step solution:
Step 1: Find the prime factors to \[32,15\& 36\]
\[32 = 2 \times 2 \times 2 \times 2 \times 2\]
\[15 = 3 \times 5\]
\[36 = 2 \times 2 \times 3 \times 3\]
Step 2: Find the common multiple of 32 and 36 we get
\[ 32 = 2 \times 2 \times 2 \times 2 \times 2 \\
  36 = 2 \times 2 \times 3 \times 3 \] [common terms 2&2 ]
 \[ = 2 \times 2 = 4\]
Step 3: find the common multiple of 15 and 36
\[
  15 = 3 \times 5 \\
  36 = 2 \times 2 \times 3 \times 3 \] [3 is the only term which is common in both]
And there are no common terms in between 32 and 15
Step 4: Now as we know that,
LCM (Least common multiple) $ = $ Product of each prime factors which are common in all three number and common in only two numbers and present only in one given number
Substituting the values from above, we get
\[LCM = 2 \times 2 \times 3 \times 2 \times 2 \times 2 \times 5 \times 3\]
Here the starting two $2$s and $3$ are common terms from \[32,15 \& 36\] and the other three $2$ s, one $5$ and one $3$ are singly occupied in \[32,15\& 36\] respectively.
Hence, we have
\[LCM = 2 \times 2 \times 3 \times 2 \times 2 \times 2 \times 5 \times 3 = 1440\]
Additional information: LCM (Least common multiple) is also equal to Product of the greatest power of each prime factor, involved in the numbers
So here we have,
\[32 = 2 \times 2 \times 2 \times 2 \times 2\]
\[15 = 3 \times 5\]
\[36 = 2 \times 2 \times 3 \times 3\]
Greatest power of prime factor \[2 = {2^5}\]
Greatest power of prime factor \[3 = {3^2}\]
Greatest power of prime factor \[5 = {5^1}\]
Now according to the definition of LCM (Least common multiple)
\[LCM = {2^5} \times {3^2} \times {5^1} = 1440\]
Hence, least common multiple of \[32,15\& 36\] is equal to \[1440\]

Note:
> Prime number is a number that is divisible only by itself and $1$(example $2$, $3$, $5$ etc).
> "Prime Factorization" is finding which prime numbers multiply together to make the original number.
> The lowest common multiple of two integers \[x\& y\] is the smallest integer which is multiple of these two numbers.