six different electronic devices beep at intervals 2,4,6,8,10,12 seconds respectively.in twenty minutes how many times these devices will beep togther
Answers
Answer:
10 times
Step-by-step explanation:
Take the L.C.M of the given intervals:
2 = 2
4 = 2^2
6 = 2*3
8 = 2^3
10 = 2*5
12 =2^2*3
L.C.M = 2^3*3*5= 120 sec i.e, 2 minutes
therefore in 20 minutes the devices will beep together = 20/2= 10 times
Concept
The LCM of any two is the value that is evenly divisible by the two given numbers. The full form of LCM is Least Common Multiple. It is also called the Least Common Divisor (LCD).
LCM is also used to add or subtract any two fractions if the denominators of the fractions are different. When performing any arithmetic operations such as addition, subtraction with fractions, LCM is used for the common denominator. This process simplifies the simplification process.
Least common multiple (LCM) is a method to find the least common multiple between any two or more numbers. A common multiple is a number that is a multiple of two or more numbers.
What is HCF?
Along with the least common multiple, you must have heard of the highest common factor (H.C.F.). HCF is used to derive the highest common factors of any two or more given integers. It is also called the greatest common divisor (GCD).
How to find LCM?
As we have already discussed, the least common multiple is the least common multiple of any two or more given numbers.
A multiple is the value we get when we multiply a number by another number. As 4 is a multiple of 2, when we multiply 2 by 2, we get 4. Similarly, in the case of a math table, you can see multiples of a number when we multiply it by 1, 2, 3, 4, 5, 6, and so on, but not with zero.
The LCM formula
Let a and b be two given integers. The formula to find the LCM for a & b is given as follows:
LCM (a,b) = (a x b)/GCD(a,b)
Where GCD (a,b) stands for Greatest Common Divisor or Greatest Common Factor of a & b.
LCM formula for fractions
The formula for finding LCM fractions is given as follows:
L.C.M. = L.C.M numerator/H.C.F denominator
Different LCM methods
There are three important methods by which we can find the LCM of two or more numbers. They are:
List of multiples
Prime factorization method
Division method
Let's learn all three methods here:
List of multiples
A method for finding the least common multiple of any given numbers is to first list the multiples of the particular numbers and then find the first common multiple between them.
LCM by prime factorization
Another method to find the LCM of given numbers is prime factorization.
LCM by division method
Finding the LCM of two numbers by division is an easy method.
First write the numbers separated by commas
Now divide the numbers by the smallest prime number.
If any number is not divisible, write down that number and continue
Keep dividing the series of numbers by primes unless we get results like 1 in the whole row
Now the LCM of the numbers will be equal to the product of all the prime numbers that we obtained by the division method
Least common multiple tree
Least common multitrees can be created using the prime number decomposition method.
LCM tree
From the tree diagram above, we can take a pair of common factors and unique factors from the branches of both numbers and multiply them as a whole to get the LCM.
Given
It is given that six different electronic devices beep at intervals 2,4,6,8,10,12 seconds respectively
Find
We need to find in twenty minutes how many times these devices will beep together
Solution
LCM of 2,4,8,10 and 12 is 120
So after each 120 seconds,they would beep together
Hence in 20 minutes, they would beep
20*60/120
= 10 times
Basically all beep at the beginning.So, total beeps together = 10+1 =11 times
Hence 11 times these devices will beep together
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