Question 990447
Selling hand towels in sets of {{{10}}} , the number of hand towels sold must be a multiple of {{{10}}} .
Selling bath towels in sets of {{{6}}} , the number of hand towels sold must be a multiple of {{{6}}} .
If the store sold the same number of each type of towel,
that same number of each type of towel sold must be a multiple of {{{10}}} , and {{{6}}} .
The smallest possible number of each type of towel sold is the least common multiple (LCM) of {{{10}}} , and {{{6}}} .
That number is {{{highlight(30)}}} .
Of course, the store may have sold
{{{2*30=60}}} of each type of towel, or
{{{3*30=90}}} , or {{{4*30=120}}} , or ...


HOW TO FIND THE LCM:
1) We could list the multiples of one of the numbers (preferably the largest).
The multiples of {{{10}}} are:
{{{10}}} , {{{20}}} , {{{30}}} , {{{40}}} , ...
Then we would check then one by one, from smallest to largest,
to see if we can find a multiple of the other number(s).
In this case, {{{10}}} , and {{{20}}} are not multiples of {{{6}}} ,
but {{{30}}} is, so {{{30}}} is the LCM of {{{10}}} , and {{{6}}} .
2) Sometimes the method above does not give you the answer fast enough.
A more general method involves finding the prime factorizations.
The prime factorizations of {{{10}}} , and {{{6}}} are
{{{10=2*5}}} and {{{6=2*3}}} .
The prime factorization of the LCM includes all prime factors found in the numbers, with the greatest exponents (if any).
In this case,
{{{LCM(10,6)=2*3*5=30}}} ,