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# The least number that is divisible by all the numbers from 1 to 10 (both inclusive) is

**(A)** 10

**(B)** 100

**(C)** 504

**(D)** 2520

Given:

Numbers between 1 and 10 (both inclusive).

To do:

We have to find the least number that is divisible by all the numbers between 1 and 10 (both inclusive).

Solution:

The LCM of 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 will be the least number that is divisible by all the numbers between 1 and 10.

__Finding LCM of all the numbers between 1 and 10 using prime factorization method__:

Writing the numbers as a product of their prime factors:

Prime factorisation of 1:

- $1\ =\ 1^1$

Prime factorisation of 2:

- $2\ =\ 2^1$

Prime factorisation of 3:

- $3\ =\ 3^1$

Prime factorisation of 4:

- $2\ \times\ 2\ =\ 2^2$

Prime factorisation of 5:

- $5\ =\ 5^1$

Prime factorisation of 6:

- $2\ \times\ 3\ =\ 2^1\ \times\ 3^1$

Prime factorisation of 7:

- $7\ =\ 7^1$

Prime factorisation of 8:

- $2\ \times\ 2\ \times\ 2\ =\ 2^3$

Prime factorisation of 9:

- $3\ \times\ 3\ =\ 3^2$

Prime factorisation of 10:

- $2\ \times\ 5\ =\ 2^1\ \times\ 5^1$

Multiplying the highest power of each prime factor:

- $1^1\ \times\ 2^3\ \times\ 3^2\ \times\ 5^1\ \times\ 7^1\ =\ 2520$

LCM(1, 2, 3, 4, 5, 6, 7, 8, 9, 10) $=$ 2520

So, the least number that is divisible by all the numbers between 1 and 10 is 2520.

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