Finite Math Examples

$20 c d$ , $40 a^{2} c^{2} d^{4}$ , $15 a b d^{3}$

Step 1

Since $20 c d, 40 a^{2} c^{2} d^{4}, 15 a b d^{3}$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $20, 40, 15$ then find LCM for the variable part $c^{1}, d^{1}, a^{2}, c^{2}, d^{4}, a^{1}, b^{1}, d^{3}$ .

Step 2

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 3

The prime factors for $20$ are $2 \cdot 2 \cdot 5$ .

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Step 3.1

$20$ has factors of $2$ and $10$ .

$2 \cdot 10$

Step 3.2

$10$ has factors of $2$ and $5$ .

$2 \cdot 2 \cdot 5$

Step 4

The prime factors for $40$ are $2 \cdot 2 \cdot 2 \cdot 5$ .

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Step 4.1

$40$ has factors of $2$ and $20$ .

$2 \cdot 20$

Step 4.2

$20$ has factors of $2$ and $10$ .

$2 \cdot 2 \cdot 10$

Step 4.3

$10$ has factors of $2$ and $5$ .

$2 \cdot 2 \cdot 2 \cdot 5$

Step 5

$15$ has factors of $3$ and $5$ .

$3 \cdot 5$

Step 6

The LCM of $20, 40, 15$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2 \cdot 2 \cdot 2 \cdot 3 \cdot 5$

Step 7

Multiply $2 \cdot 2 \cdot 2 \cdot 3 \cdot 5$ .

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Step 7.1

Multiply $2$ by $2$ .

$4 \cdot 2 \cdot 3 \cdot 5$

Step 7.2

Multiply $4$ by $2$ .

$8 \cdot 3 \cdot 5$

Step 7.3

Multiply $8$ by $3$ .

$24 \cdot 5$

Step 7.4

Multiply $24$ by $5$ .

$120$

Step 8

The factor for $c^{1}$ is $c$ itself.

$c^{1} = c$

$c$ occurs $1$ time.

Step 9

The factor for $d^{1}$ is $d$ itself.

$d^{1} = d$

$d$ occurs $1$ time.

Step 10

The factors for $a^{2}$ are $a \cdot a$ , which is $a$ multiplied by each other $2$ times.

$a^{2} = a \cdot a$

$a$ occurs $2$ times.

Step 11

The factors for $c^{2}$ are $c \cdot c$ , which is $c$ multiplied by each other $2$ times.

$c^{2} = c \cdot c$

$c$ occurs $2$ times.

Step 12

The factors for $d^{4}$ are $d \cdot d \cdot d \cdot d$ , which is $d$ multiplied by each other $4$ times.

$d^{4} = d \cdot d \cdot d \cdot d$

$d$ occurs $4$ times.

Step 13

The factor for $a^{1}$ is $a$ itself.

$a^{1} = a$

$a$ occurs $1$ time.

Step 14

The factor for $b^{1}$ is $b$ itself.

$b^{1} = b$

$b$ occurs $1$ time.

Step 15

The factors for $d^{3}$ are $d \cdot d \cdot d$ , which is $d$ multiplied by each other $3$ times.

$d^{3} = d \cdot d \cdot d$

$d$ occurs $3$ times.

Step 16

The LCM of $c^{1}, d^{1}, a^{2}, c^{2}, d^{4}, a^{1}, b^{1}, d^{3}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$c \cdot c \cdot d \cdot d \cdot d \cdot d \cdot a \cdot a \cdot b$

Step 17

Simplify $c \cdot c \cdot d \cdot d \cdot d \cdot d \cdot a \cdot a \cdot b$ .

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Step 17.1

Multiply $c$ by $c$ .

$c^{2} \cdot d \cdot d \cdot d \cdot d \cdot a \cdot a \cdot b$

Step 17.2

Multiply $d$ by $d$ by adding the exponents.

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Step 17.2.1

Move $d$ .

$c^{2} \cdot (d \cdot d) \cdot d \cdot d \cdot a \cdot a \cdot b$

Step 17.2.2

Multiply $d$ by $d$ .

$c^{2} \cdot d^{2} \cdot d \cdot d \cdot a \cdot a \cdot b$

Step 17.3

Multiply $d^{2}$ by $d$ by adding the exponents.

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Step 17.3.1

Move $d$ .

$c^{2} \cdot (d \cdot d^{2}) \cdot d \cdot a \cdot a \cdot b$

Step 17.3.2

Multiply $d$ by $d^{2}$ .

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Step 17.3.2.1

Raise $d$ to the power of $1$ .

$c^{2} \cdot (d^{1} d^{2}) \cdot d \cdot a \cdot a \cdot b$

Step 17.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$c^{2} \cdot d^{1 + 2} \cdot d \cdot a \cdot a \cdot b$

Step 17.3.3

Add $1$ and $2$ .

$c^{2} \cdot d^{3} \cdot d \cdot a \cdot a \cdot b$

Step 17.4

Multiply $d^{3}$ by $d$ by adding the exponents.

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Step 17.4.1

Move $d$ .

$c^{2} \cdot (d \cdot d^{3}) \cdot a \cdot a \cdot b$

Step 17.4.2

Multiply $d$ by $d^{3}$ .

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Step 17.4.2.1

Raise $d$ to the power of $1$ .

$c^{2} \cdot (d^{1} d^{3}) \cdot a \cdot a \cdot b$

Step 17.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$c^{2} \cdot d^{1 + 3} \cdot a \cdot a \cdot b$

Step 17.4.3

Add $1$ and $3$ .

$c^{2} \cdot d^{4} \cdot a \cdot a \cdot b$

Step 17.5

Multiply $a$ by $a$ by adding the exponents.

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Step 17.5.1

Move $a$ .

$c^{2} \cdot d^{4} \cdot (a \cdot a) \cdot b$

Step 17.5.2

Multiply $a$ by $a$ .

$c^{2} \cdot d^{4} \cdot a^{2} \cdot b$

$c^{2} d^{4} a^{2} b$

Step 18

The LCM for $20 c d, 40 a^{2} c^{2} d^{4}, 15 a b d^{3}$ is the numeric part $120$ multiplied by the variable part.

$120 c^{2} d^{4} a^{2} b$