Finding Least Number Divisible by First Ten Natural Numbers
The question asks for the least number that is exactly divisible by the first ten natural numbers. This is equivalent to finding the Least Common Multiple (LCM) of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.
Steps to Calculate LCM
- List the numbers: The first ten natural numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
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Prime Factorization: Find the prime factorization for each number:
- $1 = 1$
- $2 = 2^1$
- $3 = 3^1$
- $4 = 2^2$
- $5 = 5^1$
- $6 = 2^1 \times 3^1$
- $7 = 7^1$
- $8 = 2^3$
- $9 = 3^2$
- $10 = 2^1 \times 5^1$
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Identify Highest Powers: Identify the highest power of each unique prime factor present in the factorizations:
- Highest power of 2: $2^3$ (from 8)
- Highest power of 3: $3^2$ (from 9)
- Highest power of 5: $5^1$ (from 5 or 10)
- Highest power of 7: $7^1$ (from 7)
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Calculate LCM: Multiply these highest powers together to find the LCM.
LCM = $2^3 \times 3^2 \times 5^1 \times 7^1$
LCM = $8 \times 9 \times 5 \times 7$
LCM = $72 \times 35$
LCM = $2520$
Therefore, the least number which is divisible by the first ten natural numbers is 2520.
Similar Questions
- The H.C.F. and the L.C.M. of two numbers are 5 and 495, respectively. If one of the numbers is 55, find the other one.
- The HCF of two numbers is 16 and their LCM is 240. If one number is 48, find the other number.
- The sum of two numbers is 132 and their L.C.M. is 630. What are the two numbers?
- The LCM of the numbers 12.8 and 0.004 is:
- Find the HCF of ($3^{45} - 1$) and ($3^{35} - 1$).
- The HCF and LCM of two numbers are in the ratio of 1: 30 and the difference between the HCF and LCM is 493. Find the product of LCM and HCF.
- HCF of $\frac{1}{3}, \frac{3}{4}, \frac{4}{5}$ and $\frac{5}{6}$ is:
- Which of the following is the greatest number that, when dividing 183, 127 and 211, leaves the same remainder each time?
- If the LCM of $20x^3y^2$ and $10x^4y^4$ is $20x^4y^4$, find the HCF.
- There are four table clocks. They ring every 10 min, 15 min, 20 min and 25 min respectively. If they all ring together at 10 a.m., then at what time will they ring together again?
Important Questions from LCM and HCF
Six bells begin to toll together and toll, respectively, at intervals of 3, 4, 6, 7, 8 and 12 seconds. After how many seconds, will they toll together again?
A and B are two prime numbers such that A > B and their LCM is 209. The value of A 2 - B is:
Find the least number which when divided by 12, 18, 24 and 30 leaves 4 as remainder in each case, but when divided by 7 leaves no remainder.
Calculate the HCF of \(\frac{12}{5}\) , \(\frac{14}{15}\) and \(\frac{16}{17}\) .
Three numbers are in the proportion of 3 : 8 : 15 and their LCM is 8280. What is their HCF?
