The least Number that is divisible by all the numbers from 1 to 10 ( b...
The least Number that is divisible by all the numbers from 1 to 10 ( b...
**Explanation:**
To find the least number that is divisible by all the numbers from 1 to 10, we need to find the least common multiple (LCM) of these numbers.
**Finding the LCM of numbers 1 to 10:**
1. Prime factorize each number from 1 to 10:
- 1 = 1
- 2 = 2
- 3 = 3
- 4 = 2 * 2
- 5 = 5
- 6 = 2 * 3
- 7 = 7
- 8 = 2 * 2 * 2
- 9 = 3 * 3
- 10 = 2 * 5
2. Identify the highest power of each prime factor that appears in any of the numbers:
- 2 appears in 2, 4, 6, 8, and 10, so the highest power of 2 is 2 * 2 * 2 = 8
- 3 appears in 3, 6, and 9, so the highest power of 3 is 3 * 3 = 9
- 5 appears in 5 and 10, so the highest power of 5 is 5
3. Multiply the highest powers of each prime factor:
- 2 * 2 * 2 * 3 * 3 * 5 = 8 * 9 * 5 = 360
Therefore, the least number that is divisible by all the numbers from 1 to 10 is 360.
**Comparing the options given:**
(A) 10: This number is not divisible by 9, so it cannot be the least number divisible by all the numbers from 1 to 10.
(B) 100: This number is not divisible by 3, 4, 6, 8, or 9, so it cannot be the least number divisible by all the numbers from 1 to 10.
(C) 504: This number is not divisible by 9, so it cannot be the least number divisible by all the numbers from 1 to 10.
(D) 2520: This number is divisible by all the numbers from 1 to 10 since its prime factorization is 2 * 2 * 2 * 3 * 3 * 5 * 7. Therefore, the correct answer is option 'D'.
Hence, the least number divisible by all the numbers from 1 to 10 is 2520.
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