What is the smallest number divisible by 10 15 and 20?

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Question 19 Exercise 6.3

Answer:

Solution:

L.C.M. of 8, 15 and 20 is 120.

Prime factors of 120 = 2 x 2 x 2 x 3 x 5

Here, prime factors 2, 3 and 5 have no pair. Therefore 120 must be multiplied by 2 x 3 x 5 to make it a perfect square.

\therefore120\times2\times3\times5=3600

Hence, the smallest square number which is divisible by 8, 15 and 20 is 3600.

Video transcript

hello kids welcome to Lido Homework in this video we're going to solve this A question which says to find the smallest square number that is divisible by each of the numbers 8 15 and 20. so to find the smallest square number Divisible by 8 15 and 20 Let's find out the lcm of the given numbers 8 15 and 20. okay so let's write that down 8 15 20. two fours are eight-fifteen two tens are two twos are four fifteen as it is two fives are ten two ones are two fifteen and five as it is the next number would be three three fives are fifteen and five as it is and the next number would be five okay so the lcm is 2 into 2 into 2 into 3 into 5 this is 8 into 3 into 5 which will give you 120 so the lcm of 8 15 and 20 is 120. Now pay attention to the word square that's mentioned over here you're just not finding any smallest number that not the smallest number that is there but you find the smallest square number so for that we need to figure out if 120 is a perfect square or not so for that we'll have to check it with prime factorization let's prime factorize 120 two six are zero two threes are two are two fives are three fives are and five ones Okay, so the factors in the prime factors of 120 are 2 into 2 into 2 into 3 into 5. so when you have to find a perfect square we first group the numbers and see how many numbers don't have a group and we multiply by those numbers right so you can see here 2 has a group great now this 2 is single 3 is single and 5 is single they don't have a group so what i have to do is I'll have to multiply 120 by one more two one more three and one more five so what What I'm doing is I'm going to add one more two here, one more three here. and another five here so that i get complete groups right and then the number that i get would be a perfect square so 120 into 2 into 3 into 5 will give me 3 6 double 0. so the smallest the square number that is divisible by 8 15 and 20 is six hundred i hope i was able to uh solve your doubt with this particular video please feel free to like to share and comment and I'll see you in another video Thank you.

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LCM of 10, 15, and 20 is the smallest number among all common multiples of 10, 15, and 20. The first few multiples of 10, 15, and 20 are (10, 20, 30, 40, 50 . . .), (15, 30, 45, 60, 75 . . .), and (20, 40, 60, 80, 100 . . .) respectively. There are 3 commonly used methods to find LCM of 10, 15, 20 - by division method, by prime factorization, and by listing multiples.

What is the LCM of 10, 15, and 20?

Answer: LCM of 10, 15, and 20 is 60.

Explanation:

The LCM of three non-zero integers, a(10), b(15), and c(20), is the smallest positive integer m(60) that is divisible by a(10), b(15), and c(20) without any remainder.

Methods to Find LCM of 10, 15, and 20

Let's look at the different methods for finding the LCM of 10, 15, and 20.

• By Division Method
• By Listing Multiples
• By Prime Factorization Method

LCM of 10, 15, and 20 by Division Method

To calculate the LCM of 10, 15, and 20 by the division method, we will divide the numbers(10, 15, 20) by their prime factors (preferably common). The product of these divisors gives the LCM of 10, 15, and 20.

• Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 10, 15, and 20. Write this prime number(2) on the left of the given numbers(10, 15, and 20), separated as per the ladder arrangement.
• Step 2: If any of the given numbers (10, 15, 20) is a multiple of 2, divide it by 2 and write the quotient below it. Bring down any number that is not divisible by the prime number.
• Step 3: Continue the steps until only 1s are left in the last row.

The LCM of 10, 15, and 20 is the product of all prime numbers on the left, i.e. LCM(10, 15, 20) by division method = 2 × 2 × 3 × 5 = 60.

LCM of 10, 15, and 20 by Listing Multiples

To calculate the LCM of 10, 15, 20 by listing out the common multiples, we can follow the given below steps:

• Step 1: List a few multiples of 10 (10, 20, 30, 40, 50 . . .), 15 (15, 30, 45, 60, 75 . . .), and 20 (20, 40, 60, 80, 100 . . .).
• Step 2: The common multiples from the multiples of 10, 15, and 20 are 60, 120, . . .
• Step 3: The smallest common multiple of 10, 15, and 20 is 60.

∴ The least common multiple of 10, 15, and 20 = 60.

LCM of 10, 15, and 20 by Prime Factorization

Prime factorization of 10, 15, and 20 is (2 × 5) = 21 × 51, (3 × 5) = 31 × 51, and (2 × 2 × 5) = 22 × 51 respectively. LCM of 10, 15, and 20 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 22 × 31 × 51 = 60.
Hence, the LCM of 10, 15, and 20 by prime factorization is 60.

☛ Also Check:

LCM of 10, 15, and 20 Examples

1. Example 1: Verify the relationship between the GCD and LCM of 10, 15, and 20.

   Solution:

   The relation between GCD and LCM of 10, 15, and 20 is given as, LCM(10, 15, 20) = [(10 × 15 × 20) × GCD(10, 15, 20)]/[GCD(10, 15) × GCD(15, 20) × GCD(10, 20)]

   ⇒ Prime factorization of 10, 15 and 20:

• 10 = 21 × 51
• 15 = 31 × 51
• 20 = 22 × 51

∴ GCD of (10, 15), (15, 20), (10, 20) and (10, 15, 20) = 5, 5, 10 and 5 respectively. Now, LHS = LCM(10, 15, 20) = 60. And, RHS = [(10 × 15 × 20) × GCD(10, 15, 20)]/[GCD(10, 15) × GCD(15, 20) × GCD(10, 20)] = [(3000) × 5]/[5 × 5 × 10] = 60 LHS = RHS = 60.

Hence verified.

Show Solution >

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The LCM of 10, 15, and 20 is 60. To find the LCM (least common multiple) of 10, 15, and 20, we need to find the multiples of 10, 15, and 20 (multiples of 10 = 10, 20, 30, 40, 60 . . . .; multiples of 15 = 15, 30, 45, 60 . . . .; multiples of 20 = 20, 40, 60, 80 . . . .) and choose the smallest multiple that is exactly divisible by 10, 15, and 20, i.e., 60.

What are the Methods to Find LCM of 10, 15, 20?

The commonly used methods to find the LCM of 10, 15, 20 are:

• Listing Multiples
• Prime Factorization Method
• Division Method

Which of the following is the LCM of 10, 15, and 20? 15, 3, 60, 100

The value of LCM of 10, 15, 20 is the smallest common multiple of 10, 15, and 20. The number satisfying the given condition is 60.

What is the Least Perfect Square Divisible by 10, 15, and 20?

The least number divisible by 10, 15, and 20 = LCM(10, 15, 20)
LCM of 10, 15, and 20 = 2 × 2 × 3 × 5 [Incomplete pair(s): 3, 5]
⇒ Least perfect square divisible by each 10, 15, and 20 = LCM(10, 15, 20) × 3 × 5 = 900 [Square root of 900 = √900 = ±30]
Therefore, 900 is the required number.