Find the least number that is divisible by all the numbers between 1 and 10 (both inclusive).

Given: Numbers between 1 and 10 (both inclusive).

To find: Here 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.