Finite Math Examples

$83 \div 44$ , $268$

Step 1

To find the LCM for a list of fractions, check if denominators are similar or not.

Fractions with the same denominator:

1: $LCM (\frac{a}{b}, \frac{c}{b}) = \frac{LCM (a, c)}{b}$

Fractions with different denominators such as, $LCM (\frac{a}{b}, \frac{c}{d})$ :

1: Find the LCM of $b$ and $d = LCM (b, d)$

2: Multiply the numerator and denominator of the first fraction $\frac{a}{b}$ by $\frac{LCM (b, d)}{b}$

3: Multiply the numerator and denominator of the second fraction $\frac{c}{d}$ by $\frac{LCM (b, d)}{d}$

4: After making the denominators for all the fractions same, in this case, only two fractions, find the LCM of the new numerators

5: The LCM will be the $\frac{LCM (numerators)}{LCM (b, d)}$

Step 2

Find the LCM for the denominators of $\frac{83}{44}, 268$ .

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Step 2.1

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 2.2

The prime factors for $44$ are $2 \cdot 2 \cdot 11$ .

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Step 2.2.1

$44$ has factors of $2$ and $22$ .

$2 \cdot 22$

Step 2.2.2

$22$ has factors of $2$ and $11$ .

$2 \cdot 2 \cdot 11$

Step 2.3

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.4

The LCM of $44, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2 \cdot 2 \cdot 11$

Step 2.5

Multiply $2 \cdot 2 \cdot 11$ .

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Step 2.5.1

Multiply $2$ by $2$ .

$4 \cdot 11$

Step 2.5.2

Multiply $4$ by $11$ .

$44$

Step 3

Multiply each number by $\frac{n}{n}$ , where $n$ is a number that makes the denominator $44$ .

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Step 3.1

Divide $44$ by $44$ .

$1$

Step 3.2

Multiply the numerator and denominator of $\frac{83}{44}$ by $1$ .

$\frac{83 \cdot 1}{44 \cdot 1}$

Step 3.3

Multiply $83$ by $1$ .

$\frac{83}{44 \cdot 1}$

Step 3.4

Multiply $44$ by $1$ .

$\frac{83}{44}$

Step 3.5

Divide $44$ by $1$ .

$44$

Step 3.6

Multiply the numerator and denominator of $268$ by $44$ .

$\frac{268 \cdot 44}{1 \cdot 44}$

Step 3.7

Multiply $268$ by $44$ .

$\frac{11792}{1 \cdot 44}$

Step 3.8

Multiply $44$ by $1$ .

$\frac{11792}{44}$

Step 3.9

Write the new list with the same denominators.

$\frac{83}{44}, \frac{11792}{44}$

Step 4

Find the LCM for $83, 11792$ .

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Step 4.1

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 4.2

Since $83$ has no factors besides $1$ and $83$ .

$83$ is a prime number

Step 4.3

The prime factors for $11792$ are $2 \cdot 2 \cdot 2 \cdot 2 \cdot 11 \cdot 67$ .

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Step 4.3.1

$11792$ has factors of $2$ and $5896$ .

$2 \cdot 5896$

Step 4.3.2

$5896$ has factors of $2$ and $2948$ .

$2 \cdot 2 \cdot 2948$

Step 4.3.3

$2948$ has factors of $2$ and $1474$ .

$2 \cdot 2 \cdot 2 \cdot 1474$

Step 4.3.4

$1474$ has factors of $2$ and $737$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 737$

Step 4.3.5

$737$ has factors of $11$ and $67$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 11 \cdot 67$

Step 4.4

The LCM of $83, 11792$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 11 \cdot 67 \cdot 83$

Step 4.5

Multiply $2 \cdot 2 \cdot 2 \cdot 2 \cdot 11 \cdot 67 \cdot 83$ .

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Step 4.5.1

Multiply $2$ by $2$ .

$4 \cdot 2 \cdot 2 \cdot 11 \cdot 67 \cdot 83$

Step 4.5.2

Multiply $4$ by $2$ .

$8 \cdot 2 \cdot 11 \cdot 67 \cdot 83$

Step 4.5.3

Multiply $8$ by $2$ .

$16 \cdot 11 \cdot 67 \cdot 83$

Step 4.5.4

Multiply $16$ by $11$ .

$176 \cdot 67 \cdot 83$

Step 4.5.5

Multiply $176$ by $67$ .

$11792 \cdot 83$

Step 4.5.6

Multiply $11792$ by $83$ .

$978736$

Step 5

The answer can be found by taking the LCM of $83, 11792$ and dividing by the LCM of $44, 1$ .

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Step 5.1

Divide the LCM of $83, 11792$ by the LCM of $44, 1$ .

$\frac{978736}{44}$

Step 5.2

Divide $978736$ by $44$ .

$22244$