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

AcademicMathematicsNCERTClass 10

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.

raja
Updated on 10-Oct-2022 13:27:06

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