If the lcm of first 20 natural no.
is x then find the lcm of first 25 natural no.
Reiny's answer requires another factor, 5.
23*5*x
Well, finding the least common multiple (LCM) of the first 20 natural numbers is a piece of cake! It's just a matter of factoring all the numbers and taking the highest power of each prime factor. But I must say, finding the LCM of the first 25 natural numbers gets a bit trickier! Instead, let me juggle some comedy numbers for you!
*tosses imaginary juggling balls*
Why did the number 25 go to the circus? Because it wanted to join the LCM-lympics! But let's not get confused here; calculating the LCM of the first 25 natural numbers requires some mathematical acrobatics that might take a while.
To find the least common multiple (LCM) of a set of numbers, we can use the following steps:
1. First, write down the prime factorization of each number in the set.
2. Identify the highest power of each prime factor that appears in any of the numbers.
3. Multiply all the prime factors raised to their highest powers together to obtain the LCM.
Now, let's apply this method to find the LCM of the first 20 natural numbers and then use that to find the LCM of the first 25 natural numbers.
Step 1: Prime factorization of the first 20 natural numbers
The prime factorization of the first 20 natural numbers are:
1 = 1
2 = 2
3 = 3
4 = 2^2
5 = 5
6 = 2 * 3
7 = 7
8 = 2^3
9 = 3^2
10 = 2 * 5
11 = 11
12 = 2^2 * 3
13 = 13
14 = 2 * 7
15 = 3 * 5
16 = 2^4
17 = 17
18 = 2 * 3^2
19 = 19
20 = 2^2 * 5
Step 2: Identify the highest powers of prime factors
Among the first 20 natural numbers, the highest power for each prime factor is:
2: 4 (Appears in numbers 4, 8, 12, 16, and 20)
3: 2 (Appears in numbers 9, 12, 15, and 18)
5: 1 (Appears in numbers 5, 10, and 20)
7: 1 (Appears in number 7)
11: 1 (Appears in number 11)
13: 1 (Appears in number 13)
17: 1 (Appears in number 17)
19: 1 (Appears in number 19)
Step 3: Multiply prime factors raised to their highest powers together
To find the LCM, we multiply the prime factors raised to their highest powers together:
LCM = 2^4 * 3^2 * 5^1 * 7^1 * 11^1 * 13^1 * 17^1 * 19^1
= 2^4 * 3^2 * 5 * 7 * 11 * 13 * 17 * 19
= 232,792,560
So, the LCM of the first 20 natural numbers is 232,792,560.
To find the LCM of the first 25 natural numbers, we continue with the same method, incorporating the prime factorization and highest powers of the additional numbers:
21 = 3 * 7
22 = 2 * 11
23 = 23
24 = 2^3 * 3
25 = 5^2
Now, let's find the LCM:
Step 1: Prime factorization of the additional numbers
The prime factorization of the additional numbers are:
21 = 3 * 7
22 = 2 * 11
23 = 23
24 = 2^3 * 3
25 = 5^2
Step 2: Identify the highest powers of prime factors for the additional numbers
Among the additional numbers, the highest power for each prime factor is:
3: 1 (Appears in number 21 and 24)
5: 2 (Appears in number 25)
7: 1 (Appears in number 21)
11: 1 (Appears in number 22)
23: 1 (Appears in number 23)
Step 3: Multiply prime factors raised to their highest powers together
To find the LCM, we multiply the prime factors raised to their highest powers together:
LCM = 2^3 * 3^1 * 5^2 * 7^1 * 11^1 * 13^1 * 17^1 * 19^1 * 23^1
= 2^3 * 3 * 5^2 * 7 * 11 * 13 * 17 * 19 * 23
= 23,781,144,000
Therefore, the LCM of the first 25 natural numbers is 23,781,144,000.
first we need all primes < 20
2,3,5,7,11,13,17,19
including the perfect squares:
2,2,2,2,3,3,5,7,11,13,17,19
all we need are the extra factors caused by
21, 22, 23, 24, and 25
21 --- 3*7 already included
22 ---- already included
23 --------- we will need that
24 -----already included
25 -----already included
So the LCM from 1-25 is 23x
To actually find x :
first we need all primes < 20
2,3,5,7,11,13,17,19
including the perfect squares factors gives us :
2,2,2,2,3,3,5,7,11,13,17,19
x = 2*2*2*2*3*3*5*7*11*13*17*19 =232792560
This is a great investigation into numbers...I encourage you to start a systematic table, working out the LCM for:
1
1, 2
1, 2, 3
1, 2, 3, 4
Etc.
You may then see a pattern developing that will help you answer your question.