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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.
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