Find the LCM 5/(10^3) ; 9/(10^6)

$\frac{5}{10^{3}}$ ; $\frac{9}{10^{6}}$

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 $\frac{1}{200}, \frac{9}{1000000}$ .

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.

The prime factors for $200$ are $2 \cdot 2 \cdot 2 \cdot 5 \cdot 5$ .

$200$ has factors of $2$ and $100$ .

$2 \cdot 100$

$100$ has factors of $2$ and $50$ .

$2 \cdot 2 \cdot 50$

$50$ has factors of $2$ and $25$ .

$2 \cdot 2 \cdot 2 \cdot 25$

$25$ has factors of $5$ and $5$ .

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

The prime factors for $1000000$ are $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$ .

$1000000$ has factors of $2$ and $500000$ .

$2 \cdot 500000$

$500000$ has factors of $2$ and $250000$ .

$2 \cdot 2 \cdot 250000$

$250000$ has factors of $2$ and $125000$ .

$2 \cdot 2 \cdot 2 \cdot 125000$

$125000$ has factors of $2$ and $62500$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 62500$

$62500$ has factors of $2$ and $31250$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 31250$

$31250$ has factors of $2$ and $15625$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 15625$

$15625$ has factors of $5$ and $3125$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot 3125$

$3125$ has factors of $5$ and $625$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 625$

$625$ has factors of $5$ and $125$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5 \cdot 125$

$125$ has factors of $5$ and $25$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 25$

$25$ has factors of $5$ and $5$ .

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

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

Multiply $2$ by $2$ .

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

Multiply $4$ by $2$ .

$8 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$

Multiply $8$ by $2$ .

$16 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$

Multiply $16$ by $2$ .

$32 \cdot 2 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$

Multiply $32$ by $2$ .

$64 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$

Multiply $64$ by $5$ .

$320 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$

Multiply $320$ by $5$ .

$1600 \cdot 5 \cdot 5 \cdot 5 \cdot 5$

Multiply $1600$ by $5$ .

$8000 \cdot 5 \cdot 5 \cdot 5$

Multiply $8000$ by $5$ .

$40000 \cdot 5 \cdot 5$

Multiply $40000$ by $5$ .

$200000 \cdot 5$

Multiply $200000$ by $5$ .

$1000000$

Step 3

Multiply each number by $\frac{n}{n}$ , where $n$ is a number that makes the denominator $1000000$ .

Divide $1000000$ by $200$ .

$5000$

Multiply the numerator and denominator of $\frac{1}{200}$ by $5000$ .

$\frac{1 \cdot 5000}{200 \cdot 5000}$

Multiply $5000$ by $1$ .

$\frac{5000}{200 \cdot 5000}$

Multiply $200$ by $5000$ .

$\frac{5000}{1000000}$

Divide $1000000$ by $1000000$ .

$1$

Multiply the numerator and denominator of $\frac{9}{1000000}$ by $1$ .

$\frac{9 \cdot 1}{1000000 \cdot 1}$

Multiply $9$ by $1$ .

$\frac{9}{1000000 \cdot 1}$

Multiply $1000000$ by $1$ .

$\frac{9}{1000000}$

Write the new list with the same denominators.

$\frac{5000}{1000000}, \frac{9}{1000000}$

Step 4

Find the LCM for $5000, 9$ .

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.

The prime factors for $5000$ are $2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5 \cdot 5$ .

$5000$ has factors of $2$ and $2500$ .

$2 \cdot 2500$

$2500$ has factors of $2$ and $1250$ .

$2 \cdot 2 \cdot 1250$

$1250$ has factors of $2$ and $625$ .

$2 \cdot 2 \cdot 2 \cdot 625$

$625$ has factors of $5$ and $125$ .

$2 \cdot 2 \cdot 2 \cdot 5 \cdot 125$

$125$ has factors of $5$ and $25$ .

$2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 25$

$25$ has factors of $5$ and $5$ .

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

$9$ has factors of $3$ and $3$ .

$3 \cdot 3$

The LCM of $5000, 9$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

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

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

Multiply $2$ by $2$ .

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

Multiply $4$ by $2$ .

$8 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 5 \cdot 5$

Multiply $8$ by $3$ .

$24 \cdot 3 \cdot 5 \cdot 5 \cdot 5 \cdot 5$

Multiply $24$ by $3$ .

$72 \cdot 5 \cdot 5 \cdot 5 \cdot 5$

Multiply $72$ by $5$ .

$360 \cdot 5 \cdot 5 \cdot 5$

Multiply $360$ by $5$ .

$1800 \cdot 5 \cdot 5$

Multiply $1800$ by $5$ .

$9000 \cdot 5$

Multiply $9000$ by $5$ .

$45000$

Step 5

The answer can be found by taking the LCM of $5000, 9$ and dividing by the LCM of $200, 1000000$ .

Divide the LCM of $5000, 9$ by the LCM of $200, 1000000$ .

$\frac{45000}{1000000}$

Cancel the common factor of $45000$ and $1000000$ .

Factor $5000$ out of $45000$ .

$\frac{5000 (9)}{1000000}$

Cancel the common factors.

Factor $5000$ out of $1000000$ .

$\frac{5000 \cdot 9}{5000 \cdot 200}$

Cancel the common factor.

$\frac{5000 \cdot 9}{5000 \cdot 200}$

Rewrite the expression.

$\frac{9}{200}$

Find the LCM 5/(10^3) ; 9/(10^6)

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