9, 12, and 20 can be eliminated immediately, since none of these have 30 as a multiple [30 divided by any of them yields a remainder].

\(\displaystyle LCM(5,10) = 10\) since 10 is a multiple of 5, so 5 can be eliminated.

The set of multiples of 10 is \(\displaystyle \left \{ 10,20,\underline{30},40,50...\right \}\) and the set of multiples of 15 is \(\displaystyle \left \{ 15, \underline{30}, 45, 60,...\right \}\); since 30 is the smallest number that appears on both lists, \(\displaystyle LCM (10,15) = 30\).

\(\displaystyle \textrm{LCM }(16,20)\) is the lowest number that is a multiple of both 16 and 20, so we see which is the first number that appears in both lists of multiples.

The multiples of 16:

\(\displaystyle \left \{ 16, 32, 48, 64, \underline{80}, 96, 112,...\right \}\)

The multiples of 20:

\(\displaystyle \left \{ 20, 40, 60, \underline{80}, 100, 120,...\right \}\)

\(\displaystyle \textrm{LCM }(16,20) = 80\)

The least common multiple is the smallest number in value that is a multiple of all three of the numbers. The best way to find the least common multiple (LCM) is to make a quick list of the first few multiples of each number, and then identify the smallest number that is common to all three lists.

Multiples are numbers that you get when multiplying the original number by other numbers.

\(\displaystyle 6: 6, 12, 18, 24, 30, 36\)

\(\displaystyle 9: 9, 18, 27, 36, 45, 54\)

\(\displaystyle 12: 12, 24, 36, 48, 60, 72\)

18 is an answer choice that is common to both 6 and 9, but it is not also a multiple of 12, so it is not correct.

The smallest value that is a common multiple of all three is 36, so this is the LCM.

While 648 is a multiple of all three numbers, it is not the least common multiple of the three numbers.

This equation should be solved from left to right, finding the least common denominator for each pair. First we find the least common denominator of \(\displaystyle 3/5\) and \(\displaystyle 4/7\), which is \(\displaystyle 35\). The fractions then become \(\displaystyle 21/35\) and \(\displaystyle 20/35\). \(\displaystyle 1/3\) is then subtracted from that sum:

\(\displaystyle 3/5 + 4/7 - 1/3=21/35+20/35-1/3=63/105+60/105-35/105=88/105\)

Find all of the factors of both 9 and 6:

\(\displaystyle 9=3\times3\)

\(\displaystyle 6=2\times3\)

Eliminate all factors that 9 and 6 have in common, in this instance 3 (only eliminate the repeated instance). Multiply all factors that are not in common:

\(\displaystyle LCM=3\times2\times3=18\)

For \(\displaystyle LCM (N,15) = N\) to be a true statement, \(\displaystyle N\) must be a multiple of 15. 

\(\displaystyle 250 \div 15 = 16 \textrm{ R }10\)

\(\displaystyle 375 \div 15= 25\)

\(\displaystyle 425 \div 15 = 28 \textrm{ R }5\)

\(\displaystyle 655 \div 15 = 43 \textrm{ R }10\)

\(\displaystyle 950 \div 15 = 63 \textrm{ R } 5\)

Only 375 yields no remainder when divided by 15.

For \(\displaystyle LCM (N,20) = N\) to be a true statement, \(\displaystyle N\) must be a multiple of \(\displaystyle 20\). Each of the four values given is divisible by \(\displaystyle 20\), as seen below:

\(\displaystyle 160 \div 20 = 8\)

\(\displaystyle 320 \div 20= 16\)

\(\displaystyle 440 \div 20 = 22\)

\(\displaystyle 720 \div 20= 36\)

For \(\displaystyle LCM (N,20) = 20\) to be true, \(\displaystyle N\) must be a factor of \(\displaystyle 20\). Of the five choices, all are factors of \(\displaystyle 20\) except for \(\displaystyle 8\). This is the correct choice.

To find the least common multiple, you need to determine the multiple that both numbers share that is of the least value. List the multiples of each number and identify the first number (least value) that is in both lists:

\(\displaystyle 15: 15, 30, 45, 60, 75, 90\)

\(\displaystyle 18: 18, 36, 54, 72, 90\)

The LCM of 15 and 18 is 90 since it is the first number that shows up in both lists.

To find the least common multiple, you need to determine the multiple that all three numbers share that is of the least value. List the multiples of each number and identify the first number (least value) that is in both lists:

\(\displaystyle 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48\)

\(\displaystyle 12: 12, 24, 36, 48\)

\(\displaystyle 16: 16, 32, 48\)

The LCM of 4, 12, and 16 is 48 since it is the first number that shows up in all three lists.