Application of LCM and HCF MCQ Quiz - Objective Question with Answer for Application of LCM and HCF - Download Free PDF

Given:

Alarm 1 interval = 90 seconds

Alarm 2 interval = 44 seconds

First beep together at 6:00 PM

Formula Used:

Time for next beep together = LCM (Least Common Multiple) of the intervals

Calculation:

LCM of 90 and 44:

Prime factorization of 90 = 2 × 3² × 5

Prime factorization of 44 = 2² × 11

LCM = Product of the highest powers of all prime factors

LCM = 2² × 3² × 5 × 11 = 1980 seconds

1980 ÷ 60 = 33 minutes and 0 seconds

Next beep together = 6:00 PM + 33 minutes

⇒ Next beep together = 6:33 PM

∴ The correct answer is option 4.

Given:

Seven bells ring at intervals of 2, 3, 4, 6, 8, 9, and 12 minutes, respectively.

They started ringing simultaneously at 7:10 a.m.

Formula Used:

The time when all bells ring simultaneously is given by the Least Common Multiple (LCM) of their intervals.

Calculation:

Find the LCM of the intervals: 2, 3, 4, 6, 8, 9, and 12.

Prime factorization:

2 = 2

3 = 3

4 = 2²

6 = 2 × 3

8 = 2³

9 = 3²

12 = 2² × 3

LCM = 2³ × 3² = 8 × 9 = 72 minutes

Total time = 7:10 a.m. + 72 minutes

72 minutes = 1 hour 12 minutes

⇒ Next simultaneous ringing time = 7:10 a.m. + 1 hour 12 minutes

⇒ 8:22 a.m.

The next time when all the bells ring simultaneously is 8:22 a.m.

Given:

Time intervals for the traffic lights to turn red:

First crossing: 30 sec

Second crossing: 45 sec

Third crossing: 60 sec

Initial synchronization time: 8:30 a.m.

Concept Used:

To find the time when all lights will turn red again simultaneously, we need to find the Least Common Multiple (LCM) of the time intervals.

Calculation:

LCM of 30, 45, and 60:

Prime factorization:

30 = 2 × 3 × 5

45 = 3² × 5

60 = 2 × 2 × 3 × 5

⇒ LCM(30, 45, 60) = 180 seconds = 3 minutes

If they all turn red simultaneously at 8:30 a.m., they will turn red again simultaneously after 3 minutes.

8:30 a.m. + 3 minutes = 8:33 a.m.

The correct answer is option 1: 8:33 a.m.

Given:

The three red lights blink every 4, 6, and 8 minutes, respectively.

Interval of observation = 3 hours = 180 minutes

Formula Used:

To find how many times the lights blink together, we calculate the Least Common Multiple (LCM) of the blinking intervals and then determine how many times this LCM fits into the total interval.

Calculation:

LCM of 4, 6, and 8

Prime factorization:

4 = 2²

6 = 2 × 3

8 = 2³

LCM = 2³ × 3 = 8 × 3 = 24 minutes

Total interval = 180 minutes

Number of times they blink together = Total interval / LCM

⇒ Number of times they blink together = 180 / 24

⇒ Number of times they blink together = 7.5

Since we are excluding the starting moment we are not going to add 1 for the starting blink.

⇒ 7.5 ≈ 7 (Only consider the integer part)

⇒ 7 times

The correct answer is option 2.

Given:

Length of the rectangular courtyard = 4m 95cm = 4.95 m

Breadth of the rectangular courtyard = 16m 65cm = 16.65 m

Formula Used:

Number of tiles = Area of the courtyard / Area of one tile

Calculation:

Length = 4.95m = 495 cm

Breadth = 16.65m = 1665 cm

Now, HCF of 495 and 1665

495 = 3 × 3 × 5 × 11

1665 = 3 × 3 × 5 × 37

So, HCF = 3 × 3 × 5 = 45 cm

i.e. the side length of each square tile is 45 cm.

Now, 

the number of tiles = Area of the courtyard / Area of one tile

⇒ (495 × 1665) / (45 × 45)

⇒ 11 × 37 = 407

∴ The correct answer is option (4).

Given:

The greatest number that will divide 49, 147 and 322 to leave the same remainder in each case

Concept used:

We must now identify differences between the provided numbers.

When two numbers are divided by another number and have the same remaining, their difference must produce a remainder of zero.

Calculations:

According to the question,

The differences will be,

⇒ 147 - 49 = 98

⇒ 322 - 147 = 175

⇒ 322 - 49 = 273

The set of numbers {98,175,273}

Now HCF of {98,175,273}

⇒ 98 = 7 × 7 × 2

⇒ 175 = 5 × 5 × 7

⇒ 273 = 13 × 7 × 3

The HCF of {98,175,273} is 7

∴ The same remainder in each case is 7.

Given:

Seven bells ring at intervals of 2, 3, 4, 6, 8, 9, and 12 minutes, respectively.

They started ringing simultaneously at 7:10 a.m.

Formula Used:

The time when all bells ring simultaneously is given by the Least Common Multiple (LCM) of their intervals.

Calculation:

Find the LCM of the intervals: 2, 3, 4, 6, 8, 9, and 12.

Prime factorization:

2 = 2

3 = 3

4 = 2²

6 = 2 × 3

8 = 2³

9 = 3²

12 = 2² × 3

LCM = 2³ × 3² = 8 × 9 = 72 minutes

Total time = 7:10 a.m. + 72 minutes

72 minutes = 1 hour 12 minutes

⇒ Next simultaneous ringing time = 7:10 a.m. + 1 hour 12 minutes

⇒ 8:22 a.m.

The next time when all the bells ring simultaneously is 8:22 a.m.

Given:

Length of the rectangular courtyard = 4m 95cm = 4.95 m

Breadth of the rectangular courtyard = 16m 65cm = 16.65 m

Formula Used:

Number of tiles = Area of the courtyard / Area of one tile

Calculation:

Length = 4.95m = 495 cm

Breadth = 16.65m = 1665 cm

Now, HCF of 495 and 1665

495 = 3 × 3 × 5 × 11

1665 = 3 × 3 × 5 × 37

So, HCF = 3 × 3 × 5 = 45 cm

i.e. the side length of each square tile is 45 cm.

Now, 

the number of tiles = Area of the courtyard / Area of one tile

⇒ (495 × 1665) / (45 × 45)

⇒ 11 × 37 = 407

∴ The correct answer is option (4).

Given:

The greatest number that will divide 49, 147 and 322 to leave the same remainder in each case

Concept used:

We must now identify differences between the provided numbers.

When two numbers are divided by another number and have the same remaining, their difference must produce a remainder of zero.

Calculations:

According to the question,

The differences will be,

⇒ 147 - 49 = 98

⇒ 322 - 147 = 175

⇒ 322 - 49 = 273

The set of numbers {98,175,273}

Now HCF of {98,175,273}

⇒ 98 = 7 × 7 × 2

⇒ 175 = 5 × 5 × 7

⇒ 273 = 13 × 7 × 3

The HCF of {98,175,273} is 7

∴ The same remainder in each case is 7.

Given:

Time intervals for the traffic lights to turn red:

First crossing: 30 sec

Second crossing: 45 sec

Third crossing: 60 sec

Initial synchronization time: 8:30 a.m.

Concept Used:

To find the time when all lights will turn red again simultaneously, we need to find the Least Common Multiple (LCM) of the time intervals.

Calculation:

LCM of 30, 45, and 60:

Prime factorization:

30 = 2 × 3 × 5

45 = 3² × 5

60 = 2 × 2 × 3 × 5

⇒ LCM(30, 45, 60) = 180 seconds = 3 minutes

If they all turn red simultaneously at 8:30 a.m., they will turn red again simultaneously after 3 minutes.

8:30 a.m. + 3 minutes = 8:33 a.m.

The correct answer is option 1: 8:33 a.m.

Given:

Seven bells ring at intervals of 2, 3, 4, 6, 8, 9, and 12 minutes, respectively.

They started ringing simultaneously at 7:10 a.m.

Formula Used:

The time when all bells ring simultaneously is given by the Least Common Multiple (LCM) of their intervals.

Calculation:

Find the LCM of the intervals: 2, 3, 4, 6, 8, 9, and 12.

Prime factorization:

2 = 2

3 = 3

4 = 2²

6 = 2 × 3

8 = 2³

9 = 3²

12 = 2² × 3

LCM = 2³ × 3² = 8 × 9 = 72 minutes

Total time = 7:10 a.m. + 72 minutes

72 minutes = 1 hour 12 minutes

⇒ Next simultaneous ringing time = 7:10 a.m. + 1 hour 12 minutes

⇒ 8:22 a.m.

The next time when all the bells ring simultaneously is 8:22 a.m.

Given:

The three red lights blink every 4, 6, and 8 minutes, respectively.

Interval of observation = 3 hours = 180 minutes

Formula Used:

To find how many times the lights blink together, we calculate the Least Common Multiple (LCM) of the blinking intervals and then determine how many times this LCM fits into the total interval.

Calculation:

LCM of 4, 6, and 8

Prime factorization:

4 = 2²

6 = 2 × 3

8 = 2³

LCM = 2³ × 3 = 8 × 3 = 24 minutes

Total interval = 180 minutes

Number of times they blink together = Total interval / LCM

⇒ Number of times they blink together = 180 / 24

⇒ Number of times they blink together = 7.5

Since we are excluding the starting moment we are not going to add 1 for the starting blink.

⇒ 7.5 ≈ 7 (Only consider the integer part)

⇒ 7 times

The correct answer is option 2.

Given:

Alarm 1 interval = 90 seconds

Alarm 2 interval = 44 seconds

First beep together at 6:00 PM

Formula Used:

Time for next beep together = LCM (Least Common Multiple) of the intervals

Calculation:

LCM of 90 and 44:

Prime factorization of 90 = 2 × 3² × 5

Prime factorization of 44 = 2² × 11

LCM = Product of the highest powers of all prime factors

LCM = 2² × 3² × 5 × 11 = 1980 seconds

1980 ÷ 60 = 33 minutes and 0 seconds

Next beep together = 6:00 PM + 33 minutes

⇒ Next beep together = 6:33 PM

∴ The correct answer is option 4.