# LCM and GCM Word Problems

### Solutions

1A beacon flashes its light every 12 seconds, another every 18 seconds and a third every minute. At 6.30 pm the three flash simultaneously.

Find out the times when the three flash simultaneously again in the next five minutes.

2A businessman goes to Chicago every 18 days for one day and another businessman every 24 days, also for only one day. Today, both men are in Chicago.

Within how many days will the two business men be in Chicago again at the same time?

3What is the smallest number that when divided in separate occaisons, 15, 20, 36 and 48, gives the remainder of 9 in every case?

4There are 3 differently sized casks of wine in a cellar whose capacities are: 250 liters, 360 litres, and 540 litres. The owner of the cellar wants to package the wine in barrels with an equal amount of wine in each one. Calculate the maximum capacities of these barrels so that the owner can package equal amounts of wine in each cask, and determine the quantity of barrels he will need.

5The floor of a room that needs to be tiled is 5 m long and 3 m wide.

Determine the ideal size of the tiles and the number of the tiles needed, such that the number of tiles that are placed is minimal and none of them are to be cut. Keep in mind that all tiles are to be the same size.

6A trader wants to put 12,028 apples and 12,772 oranges into boxes. Each box is to contain an equal number of apples and an equal number of oranges and also the greatest number of each. Find the ideal number of oranges and apples for each box and the number of boxes needed.

7What is the size of the largest possible square tile that can fit an in a room 8 m long and 6.4 meters wide without being cut? How many tiles are needed?

## 1

A beacon flashes its light every 12 seconds, another every 18 seconds and a third every minute. At 6.30 pm the three flash simultaneously.

Find out the times when the three flash simultaneously again in the next five minutes.

12 = 2^{2} · 3

18 = 2 · 3^{2}

60 = 2^{2} · 3 · 5

LCM (12, 18, 60) = 2^{2} · 3^{2} · 5= 180

180 : 60 = 3

Only at 6.33 pm..

## 2

A businessman goes to Chicago every 18 days for one day and another businessman every 24 days, also for only one day. Today, both men are in Chicago.

Within how many days will the two business men be in Chicago again at the same time?

18 = 2 · 3^{2}

24 = 2^{3} · 3

LCM (18, 24) = 2^{3} · 3^{2} = 72

Within 72 days.

## 3

What is the smallest number that when divided in separate occaisons, 15, 20, 36 and 48 , gives the remainder of 9 in every case?

LCM (15, 20, 36, 48) = 2^{4} · 3^{2} · 5 = 720

720 + 9 = 729

## 4

There are 3 differently sized casks of wine in a cellar whose capacities are: 250 liters, 360 litres, and 540 litres. The owner of the cellar wants to package the wine in barrels with an equal amount of wine in each one. Calculate the maximum capacities of these barrels so that the owner can package equal amounts of wine in each cask, and determine the quantity of barrels he will need.

GCD (250, 360, 540) = 10

Capacity of the barrels= 10 l.

Number of barrels of C_{1} = 250/10 = 25

Number of barrels of C_{2} = 360/10 = 36

Number of barrels of C_{3} = 540/10 = 54

Number of barrels = 25 + 36 + 54 = 115 barrels.

## 5

5The floor of a room that needs to be tiled is 5 m long and 3 m wide.

Determine the ideal size of the tiles and the number of the tiles needed, such that the number of tiles that are placed is minimal and none of them are to be cut. Keep in mind that all tiles are to be the same size.

3 m = 30 dm 30 = 2 · 3 · 5

5 m = 50 dm 50 = 2 · 5^{ 2 }

A = 30 · 50 = 1,500 dm^{2 }

G.C.D. (30, 50) = 2 · 5= 10 dm aside

A_{b } = 10^{2} = 100 dm^{ 2 }

1,500 dm^{2} : 100 dm^{2 } = 15 tiles

## 6

A trader wants to put 12,028 apples and 12,772 oranges into boxes. Each box is to contain an equal number of apples and an equal number of oranges and also the greatest number of each. Find the ideal number of oranges and apples for each box and the number of boxes needed.

GCD (12,028, 12,772) = 124

124 oranges in each box.

Boxes of oranges = 12,772/124 = 103

Boxes of apples = 12,028/124 = 97

Boxes necessary = 103 + 97 = 200

## 7

What is the size of the largest possible square tile that can fit an in a room 8 m long and 6.4 meters wide without being cut? How many tiles are needed?

8 m = 80 dm 80 = 2^{4} · 5

6.4 m = 64 dm 64 = 2^{6 }

GCD (80, 64) = 2^{4} = 16 dm aside

A b = 16^{2} = 256 dm^{2 }

A = 80 · 64 = 5,120 dm^{2 }

5,120 dm^{2 } : 256 dm^{2 } = 20 tiles