The least common multiple (LCM) of two numbers is the smallest whole number that is a multiple of both numbers. Let's see a few examples!

Numbers	Multiples of Numbers	Common Multiples	Least Common Multiple
$2$ and $3$	$Multiples of 2 : Multiples of 3 : 2, 4, 6, 8, 10, 12, \dots 3, 6, 9, 12, 15, \dots$	$\underline{6}, 12, 18, 24, \dots$	$LCM (2, 3) = 6$
$8$ and $12$	$Multiples of 8 : Multiples of 12 : 8, 16, 24, 32, 40, 48, \dots 12, 24, 36, 48, \dots$	$\underline{24}, 48, 72, 96, \dots$	$LCM (8, 12) = 24$

We want to find three numbers that have a least common multiple of

100 .

To do so, recall that a factor pair is a combination of two factors that can be multiplied together to equal a number. Let's find the factor pairs of

100

by listing its factors!

Factors of 100 : 1, 2, 4, 5, 10, 20, 25, 50, 100

Now, we can write the factor pairs of each number by finding the product of two factors that equals

100 .

Number	Product of Two Factors	Factor Pairs
$100$	$1 \times 100 = 100 02 \times 50 = 100 04 \times 25 = 100 05 \times 20 = 100 10 \times 10 = 100$	$(1, 100) (2, 50) (4, 25) (5, 20) (10, 10)$

Remember that when a number is multiplied by an integer, the result is a multiple of the first number.

Note that the above products have the same form as the products we used to find the factor pairs. Then, we can list these products to identify the numbers that have

100

as a multiple. Let's do it!

1 \times 100 100 \times 1 2 \times 50 50 \times 2 4 \times 25 25 \times 4 5 \times 20 20 \times 5 10 \times 10 = 100 \to 100 is multiple of 1 = 100 \to 100 is multiple of 100 = 100 \to 100 is multiple of 2 = 100 \to 100 is multiple of 50 = 100 \to 100 is multiple of 4 = 100 \to 100 is multiple of 25 = 100 \to 100 is multiple of 5 = 100 \to 100 is multiple of 20 = 100 \to 100 is multiple of 10

Now we can list the multiples of the numbers that have

100

as a factor to find three numbers with an LCM equal to

100 .

Let's ignore the

1

and

100

because they are the identities.

Multiples of 2 Multiples of 4 Multiples of 5 Multiples of 10 Multiples of 20 Multiples of 25 Multiples of 50 : 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, . . . : 4, 8, 12, 16, 20, 24, 28, 32, 36, . . . : 5, 10, 15, . . ., 95, 100, 105, 110, . . . : 10, 20, 30, . . ., 70, 80, 90, 100, . . . : 20, 40, 60, 80, 100, 120, 140, . . . : 25, 50, 75, 100, 125, 150, 175, . . . : 50, 100, 150, 200, 250, 300, 350, . . .

From the above list, we can see that

5,

20,

and

25

have an LCM of

100 .

Note that there are other three-number sets we could create out of this list that have an LCM of

100 .

Exercise