Question

The least number that is divisible by all the numbers from 1 to 8 (both inclusive) is​

Answer 1

Step-by-step explanation:

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The least number divisible by all the numbers from 1 to 10 will be the LCM of these numbers.

We have,

1 = 1

2 = 2 × 1

3 = 3 × 1

4 = 2 × 2

5 = 5 × 1

6 = 2 × 3

7 = 7 × 1

8 = 2 × 2 × 2

9 = 3 × 3

10 = 2 × 5

So, LCM of these numbers = 1 × 2 × 2 × 2 × 3 × 3 × 5 × 7 = 2520

Hence, least number divisible by all the numbers from 1 to 10 is 2520

Answer 2

The least number that is divisible by 1 and 8(both inclusive) is 840.

Solution: The least number that is divisible by 1 to 8 should be the LCM of every number between 1 to 8.

For finding the LCM of these numbers, we write the prime factorisation of each of these numbers.

Prime factorisation of 1 = 1

Prime factorisation of 2= 2

Prime factorisation of 3 = 3

Prime factorisation of 4 = 2×2

$= {2}^{2}$

Prime factorisation of 5 = 5

Prime factorisation of 6 = 2×3

Prime factorisation of 7 = 7

Prime factorisation of 8 = 2×2×2

$= {2}^{3}$

The LCM is the product of the highest power of each prime factors.

The highest powers of prime factors 2,3,5 and 7 are 3,1,1 and 1 respectively in these numbers.

Therefore, LCM

$= {2}^{3} \times {3}^{1} \times {5}^{1} \times {7}^{1}$

= 840