if the LCM of first 20 natural numbers is X, then what will be the LCM of first 25 natral numbers

since 23 is prime, it will be at least X*23

Since 25 = 5^2, the value will be X*23*5
because no other n<25 has 5 as a factor twice

All the factors of 21,22,24 are already in X

To find the LCM (Least Common Multiple) of the first 25 natural numbers, you can first calculate the LCM of the first 20 natural numbers and then find the additional factors from 21 to 25.

Step 1: Find the LCM of the first 20 natural numbers.
To find the LCM of the first 20 natural numbers, you can multiply the prime factors of each number and take the highest power of each prime factor.

Prime factorization of the first 20 natural numbers:
1 = 1
2 = 2^1
3 = 3^1
4 = 2^2
5 = 5^1
6 = 2^1 * 3^1
7 = 7^1
8 = 2^3
9 = 3^2
10 = 2^1 * 5^1
11 = 11^1
12 = 2^2 * 3^1
13 = 13^1
14 = 2^1 * 7^1
15 = 3^1 * 5^1
16 = 2^4
17 = 17^1
18 = 2^1 * 3^2
19 = 19^1
20 = 2^2 * 5^1

Take the highest power of each prime factor:
2^4, 3^2, 5^1, 7^1, 11^1, 13^1, 17^1, 19^1

Multiply these prime factors together:
2^4 * 3^2 * 5^1 * 7^1 * 11^1 * 13^1 * 17^1 * 19^1 = 232,792,560

Therefore, the LCM of the first 20 natural numbers, X, is 232,792,560.

Step 2: Find the additional factors from 21 to 25.
The additional factors from 21 to 25 are 21, 22, 23, 24, and 25.

Prime factorization of 21 = 3^1 * 7^1.
Prime factorization of 22 = 2^1 * 11^1.
Prime factorization of 23 = 23^1.
Prime factorization of 24 = 2^3 * 3^1.
Prime factorization of 25 = 5^2.

Take the highest power of each prime factor:
3^1, 7^1, 2^3, 11^1, 23^1, 5^2.

Multiply these prime factors together:
3^1 * 7^1 * 2^3 * 11^1 * 23^1 * 5^2 = 2,090,275

Step 3: Calculate the LCM of the first 25 natural numbers.
To find the LCM of the first 25 natural numbers, multiply the LCM of the first 20 natural numbers (X) by the additional factors calculated in step 2.

LCM of the first 25 natural numbers = X * 2,090,275

LCM of the first 25 natural numbers = 232,792,560 * 2,090,275

LCM of the first 25 natural numbers = 486,580,401,635,000

Therefore, the LCM of the first 25 natural numbers is 486,580,401,635,000.

To find the LCM (Least Common Multiple) of any set of numbers, we need to find the smallest number that is divisible by each of the given numbers.

In this case, we want to find the LCM of the first 25 natural numbers, given that the LCM of the first 20 natural numbers is X.

To solve this, we need to understand the concept of prime factorization.

Step 1: Prime factorization of the LCM of the first 20 natural numbers (X)
- Let's break down the numbers from 1 to 20 into their prime factors:
- 1: Prime factorization is 1
- 2: Prime factorization is 2
- 3: Prime factorization is 3
- 4: Prime factorization is 2^2 (2 raised to the power of 2)
- 5: Prime factorization is 5
- 6: Prime factorization is 2 * 3
- 7: Prime factorization is 7
- 8: Prime factorization is 2^3 (2 raised to the power of 3)
- 9: Prime factorization is 3^2 (3 raised to the power of 2)
- 10: Prime factorization is 2 * 5
- 11: Prime factorization is 11
- 12: Prime factorization is 2^2 * 3
- 13: Prime factorization is 13
- 14: Prime factorization is 2 * 7
- 15: Prime factorization is 3 * 5
- 16: Prime factorization is 2^4 (2 raised to the power of 4)
- 17: Prime factorization is 17
- 18: Prime factorization is 2 * 3^2
- 19: Prime factorization is 19
- 20: Prime factorization is 2^2 * 5

Now, we can calculate the LCM by including the maximum powers of each prime number that appear in the prime factorization.

Step 2: Calculate the LCM of first 25 natural numbers using the prime factorization of X
- In the prime factorization of X, we have the maximum powers of prime numbers up to 20.
- Now, for the numbers from 21 to 25, let's determine their prime factorizations:
- 21: Prime factorization is 3 * 7
- 22: Prime factorization is 2 * 11
- 23: Prime factorization is 23
- 24: Prime factorization is 2^3 * 3
- 25: Prime factorization is 5^2

Step 3: Calculate the LCM of first 25 natural numbers
- To calculate the LCM, we include the maximum powers of each prime number that appear in the prime factorization of numbers up to 25:
- Prime factorization of 21: 3 * 7 (No new prime powers added)
- Prime factorization of 22: 2 * 11 (No new prime powers added)
- Prime factorization of 23: 23 (No new prime powers added)
- Prime factorization of 24: 2^3 * 3 (The power of 2 increased from 2^2 to 2^3)
- Prime factorization of 25: 5^2 (No new prime powers added)

Now, we can calculate the LCM with the maximum powers of each prime number:
LCM = (2^3) * (3^1) * (5^2) * (7^1) * (11^1) * (13^1) * (17^1) * (19^1) * (23^1)

Therefore, the LCM of the first 25 natural numbers is (2^3) * (3^1) * (5^2) * (7^1) * (11^1) * (13^1) * (17^1) * (19^1) * (23^1).

You idiot