Question 9

What is the least number that is divisible by all the numbers $1$ to $10$

Solution:

To find the least number that is divisible by all the numbers from 1 to 10, we need to determine the Least Common Multiple (LCM) of these numbers. Here’s a step-by-step solution:

Step 1: List the numbers from 1 to 10
The numbers we need to consider are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Step 2: Prime Factorization of each number
We will find the prime factorization of each number:
– 1: 1 (not a prime)
– 2: 2
– 3: 3
– 4: 2 × 2 (or $$2^2$$)
– 5: 5
– 6: 2 × 3
– 7: 7
– 8: 2 × 2 × 2 (or $$2^3$$)
– 9: 3 × 3 (or $$3^2$$)
– 10: 2 × 5

Step 3: Identify the highest powers of each prime factor
Now, we need to take the highest power of each prime number that appears in the factorizations:
– 2: Highest power is $$2^3$$ (from 8)
– 3: Highest power is $$3^2$$ (from 9)
– 5: Highest power is $$5^1$$ (from 5 and 10)
– 7: Highest power is $$7^1$$ (from 7)

Step 4: Calculate the LCM
Now, we can calculate the LCM by multiplying these highest powers together:
$$\text{LCM}=2^3\times 3^2\times 5^1\times 7^1$$

Calculating this step-by-step:
1. $$2^3=8$$
2. $$3^2=9$$
3. $$5^1=5$$
4. $$7^1=7$$

Now multiply these values:
– First, calculate $$8\times 9=72$$
– Next, calculate $$72\times 5=360$$
– Finally, calculate $$360\times 7=2520$$

Thus, the least number that is divisible by all the numbers from 1 to 10 is 2520.

Final Answer:
The least number that is divisible by all the numbers from 1 to 10 is 2520.