A number is divisible by 5 and 6. It may not be divisible by (A) 10 (B) 15 (C) 30 (D) 60?
Answers
Answer:
The LCM of 6 and 5 is 30. So, 30 is divisible by 10, 15 and 30 in the given options. But, 30 is not divisible by 60.
Step-by-step explanation:
Any number divisible by 5 and 6 will be either 30 or multiple of 30, but 30 is not divisible by 60.
Example:
(i) 90 is divisible by 5 and 6. It is also divisible by 15 and 30, but not divisible by 60
(ii) Similarly, 270 is divisible by 5,6,15 and 30 but not 60.
Hence option D is the correct answer.
LCM denotes the least common factor or multiple of any two or more given integers. For example, L.C.M of 16 and 20 will be 2 x 2 x 2 x 2 x 5 = 80, where 80 is the smallest common multiple for numbers 16 and 20.
Now, if we consider the multiples of 16 and 20, we get;
16 → 16, 32, 48, 64, 80,…
20 → 20, 40, 60, 80,…,
We can see that the first common multiple for both numbers is 80. This proves the method of LCM as correct.
Also check: LCM of two numbers
What is HCF?
Along with the least common multiple, you must have heard about 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 as Greatest Common Divisor (GCD).
For example, the H.C.F. of 2,6,8 is 2, because all the three numbers can be divided with the factor 2, commonly. H.C.F. and L.C.M. both have equal importance in Maths.
Properties of LCM
Properties Description
Associative property LCM(a, b) = LCM(b, a)
Commutative property LCM(a, b, c) = LCM(LCM(a, b), c) = LCM(a, LCM(b, c))
Distributive property LCM(da, db, dc) = dLCM(a, b, c)
How to Find LCM?
As we have already discussed, the least common multiple is the smallest common multiple for any two or more given numbers.
A multiple is a value we get when we multiply a number with another number. Like 4 is a multiple of 2, as we multiply 2 with 2, we get 4. Similarly, in the case of the maths table, you can see the multiples of a number when we multiply them from 1, 2, 3, 4, 5, 6 and so on but not with zero.
LCM Formula
Let a and b are two given integers. The formula to find the LCM of a & b is given by:
LCM (a,b) = (a x b)/GCD(a,b)
Where GCD (a,b) means Greatest Common Divisor or Highest Common Factor of a & b.
LCM Formula for Fractions
The formula to find the LCM of fractions is given by:
L.C.M. = L.C.M Of Numerator/H.C.F Of Denominator
Different Methods of LCM
There are three important methods by which we can find the LCM of two or more numbers. They are:
Listing the Multiples
Prime Factorisation Method
Division Method
Let us learn here all three methods:
Listing the Multiples
The method to find the least common multiple of any given numbers is first to list down the multiples of specific numbers and then find the first common multiple between them.
Suppose there are two numbers 11 and 33. Then by listing the multiples of 11 and 33, we get;
Multiples of 11 = 11, 22, 33, 44, 55, ….
Multiples of 33 = 33, 66, 99, ….
We can see, the first common multiple or the least common multiple of both the numbers is 33. Hence, the LCM (11, 33) = 33.
LCM By Prime Factorisation
Another method to find the LCM of the given numbers is prime factorization. Suppose, there are three numbers 12, 16 and 24. Let us write the prime factors of all three numbers individually.
12 = 2 x 2 x 3
16 = 2 x 2 x 2 x 2
24 = 2 x 2 x 2 x 3
Now writing the prime factors of all the three numbers together, we get;
12 x 16 x 24 = 2 x 2 x 3 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3
Now pairing the common prime factors we get the LCM. Hence, there are four 2’s and one 3. So the LCM of 12, 16 and 24 will be;
LCM (12, 16, 24) = 2 x 2 x 2 x 2 x 3 = 48
LCM By Division Method
Finding LCM of two numbers by division method is an easy method. Below are the steps to find the LCM by division method:
First, write the numbers, separated by commas
Now divide the numbers, by the smallest prime number.
If any number is not divisible, then write down that number and proceed further
Keep on dividing the row of numbers by prime numbers, unless we get the results as 1 in the complete row
Now LCM of the numbers will be equal to the product of all the prime numbers we obtained in the division method
Let us understand with the help of examples.
Example: Find LCM of 10, 18 and 20 by division method.
Solution: Let us draw a table to divide the numbers by prime factors.
Prime factors 1st number 2nd number 3rd number
2 10 18 20
2 5 9 10
3 5 9 5
3 5 3 5
5 5 1 5
1 1 1
Therefore, LCM (10, 18, 20) = 2 x 2 x 3 x 3 x 5 = 180
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