Finite Math Examples

419

$419$ ,

290

$290$ ,

195

$195$ ,

155

$155$ ,

283

$283$ ,

330

$330$ ,

441

$441$ ,

200

$200$

Step 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

Since

419

$419$ has no factors besides

1

$1$ and

419

$419$ .

419

$419$ is a prime number

Step 3

The prime factors for

290

$290$ are

2 \cdot 5 \cdot 29

$2 \cdot 5 \cdot 29$ .

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Step 3.1

290

$290$ has factors of

2

$2$ and

145

$145$ .

2 \cdot 145

$2 \cdot 145$

Step 3.2

145

$145$ has factors of

5

$5$ and

29

$29$ .

2 \cdot 5 \cdot 29

$2 \cdot 5 \cdot 29$

2 \cdot 5 \cdot 29

$2 \cdot 5 \cdot 29$

Step 4

The prime factors for

195

$195$ are

3 \cdot 5 \cdot 13

$3 \cdot 5 \cdot 13$ .

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Step 4.1

195

$195$ has factors of

3

$3$ and

65

$65$ .

3 \cdot 65

$3 \cdot 65$

Step 4.2

65

$65$ has factors of

5

$5$ and

13

$13$ .

3 \cdot 5 \cdot 13

$3 \cdot 5 \cdot 13$

3 \cdot 5 \cdot 13

$3 \cdot 5 \cdot 13$

Step 5

155

$155$ has factors of

5

$5$ and

31

$31$ .

5 \cdot 31

$5 \cdot 31$

Step 6

Since

283

$283$ has no factors besides

1

$1$ and

283

$283$ .

283

$283$ is a prime number

Step 7

The prime factors for

330

$330$ are

2 \cdot 3 \cdot 5 \cdot 11

$2 \cdot 3 \cdot 5 \cdot 11$ .

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Step 7.1

330

$330$ has factors of

2

$2$ and

165

$165$ .

2 \cdot 165

$2 \cdot 165$

Step 7.2

165

$165$ has factors of

3

$3$ and

55

$55$ .

2 \cdot 3 \cdot 55

$2 \cdot 3 \cdot 55$

Step 7.3

55

$55$ has factors of

5

$5$ and

11

$11$ .

2 \cdot 3 \cdot 5 \cdot 11

$2 \cdot 3 \cdot 5 \cdot 11$

2 \cdot 3 \cdot 5 \cdot 11

$2 \cdot 3 \cdot 5 \cdot 11$

Step 8

The prime factors for

441

$441$ are

3 \cdot 3 \cdot 7 \cdot 7

$3 \cdot 3 \cdot 7 \cdot 7$ .

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Step 8.1

441

$441$ has factors of

3

$3$ and

147

$147$ .

3 \cdot 147

$3 \cdot 147$

Step 8.2

147

$147$ has factors of

3

$3$ and

49

$49$ .

3 \cdot 3 \cdot 49

$3 \cdot 3 \cdot 49$

Step 8.3

49

$49$ has factors of

7

$7$ and

7

$7$ .

3 \cdot 3 \cdot 7 \cdot 7

$3 \cdot 3 \cdot 7 \cdot 7$

3 \cdot 3 \cdot 7 \cdot 7

$3 \cdot 3 \cdot 7 \cdot 7$

Step 9

The prime factors for

200

$200$ are

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

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

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Step 9.1

200

$200$ has factors of

2

$2$ and

100

$100$ .

2 \cdot 100

$2 \cdot 100$

Step 9.2

100

$100$ has factors of

2

$2$ and

50

$50$ .

2 \cdot 2 \cdot 50

$2 \cdot 2 \cdot 50$

Step 9.3

50

$50$ has factors of

2

$2$ and

25

$25$ .

2 \cdot 2 \cdot 2 \cdot 25

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

Step 9.4

25

$25$ has factors of

5

$5$ and

5

$5$ .

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

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

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

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

Step 10

The LCM of

419, 290, 195, 155, 283, 330, 441, 200

$419, 290, 195, 155, 283, 330, 441, 200$ 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 7 \cdot 7 \cdot 11 \cdot 13 \cdot 29 \cdot 31 \cdot 283 \cdot 419

$2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 7 \cdot 7 \cdot 11 \cdot 13 \cdot 29 \cdot 31 \cdot 283 \cdot 419$

Step 11

Multiply

$2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 7 \cdot 7 \cdot 11 \cdot 13 \cdot 29 \cdot 31 \cdot 283 \cdot 419$ .

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Step 11.1

Multiply

$2$ by

$2$ .

$4 \cdot 2 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 7 \cdot 7 \cdot 11 \cdot 13 \cdot 29 \cdot 31 \cdot 283 \cdot 419$

Step 11.2

Multiply

$4$ by

$2$ .

$8 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdot 7 \cdot 7 \cdot 11 \cdot 13 \cdot 29 \cdot 31 \cdot 283 \cdot 419$

Step 11.3

Multiply

$8$ by

$3$ .

$24 \cdot 3 \cdot 5 \cdot 5 \cdot 7 \cdot 7 \cdot 11 \cdot 13 \cdot 29 \cdot 31 \cdot 283 \cdot 419$

Step 11.4

Multiply

$24$ by

$3$ .

$72 \cdot 5 \cdot 5 \cdot 7 \cdot 7 \cdot 11 \cdot 13 \cdot 29 \cdot 31 \cdot 283 \cdot 419$

Step 11.5

Multiply

$72$ by

$5$ .

$360 \cdot 5 \cdot 7 \cdot 7 \cdot 11 \cdot 13 \cdot 29 \cdot 31 \cdot 283 \cdot 419$

Step 11.6

Multiply

$360$ by

$5$ .

$1800 \cdot 7 \cdot 7 \cdot 11 \cdot 13 \cdot 29 \cdot 31 \cdot 283 \cdot 419$

Step 11.7

Multiply

$1800$ by

$7$ .

$12600 \cdot 7 \cdot 11 \cdot 13 \cdot 29 \cdot 31 \cdot 283 \cdot 419$

Step 11.8

Multiply

$12600$ by

$7$ .

$88200 \cdot 11 \cdot 13 \cdot 29 \cdot 31 \cdot 283 \cdot 419$

Step 11.9

Multiply

$88200$ by

$11$ .

$970200 \cdot 13 \cdot 29 \cdot 31 \cdot 283 \cdot 419$

Step 11.10

Multiply

$970200$ by

$13$ .

$12612600 \cdot 29 \cdot 31 \cdot 283 \cdot 419$

Step 11.11

Multiply

$12612600$ by

$29$ .

$365765400 \cdot 31 \cdot 283 \cdot 419$

Step 11.12

Multiply

$365765400$ by

$31$ .

$11338727400 \cdot 283 \cdot 419$

Step 11.13

Multiply

$11338727400$ by

$283$ .

$3208859854200 \cdot 419$

Step 11.14

Multiply

$3208859854200$ by

$419$ .

$1344512278909800$